Documentation Verification Report

IntermediateValue

πŸ“ Source: Mathlib/Topology/Order/IntermediateValue.lean

Statistics

MetricCount
Definitions0
TheoremsstrictAnti_of_inj_boundedOrder, strictMonoOn_of_inj_rigidity, strictMono_of_inj, strictMono_of_inj_boundedOrder, strictMono_of_inj_boundedOrder', surjective, surjective', strictAntiOn_of_injOn_Icc, strictMonoOn_of_injOn_Icc, strictMonoOn_of_injOn_Icc', strictMonoOn_of_injOn_Ioo, surjOn_Icc, surjOn_of_tendsto, surjOn_of_tendsto', surjOn_uIcc, Icc_subset_of_forall_exists_gt, Icc_subset_of_forall_exists_lt, Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset, Icc_subset_of_forall_mem_nhdsWithin, mem_of_ge_of_forall_exists_gt, mem_of_ge_of_forall_exists_lt, Icc_subset, Ioo_csInf_csSup_subset, Icc_subset, Iio_csSup_subset, Ioi_csInf_subset, eq_univ_of_unbounded, intermediate_value, intermediate_value_Ici, intermediate_value_Ico, intermediate_value_Iic, intermediate_value_Iii, intermediate_value_Iio, intermediate_value_Ioc, intermediate_value_Ioi, intermediate_value_Ioo, intermediate_valueβ‚‚, intermediate_valueβ‚‚_eventually₁, intermediate_valueβ‚‚_eventuallyβ‚‚, mem_intervals, ordConnected, isPreconnected, eq_Icc_csInf_csSup_of_connected_bdd_closed, exists_mem_Icc_isFixedPt, exists_mem_Icc_isFixedPt_of_mapsTo, exists_mem_uIcc_isFixedPt, intermediate_value_Icc, intermediate_value_Icc', intermediate_value_Ico, intermediate_value_Ico', intermediate_value_Ioc, intermediate_value_Ioc', intermediate_value_Ioo, intermediate_value_Ioo', intermediate_value_uIcc, intermediate_value_univ, intermediate_value_univβ‚‚, intermediate_value_univβ‚‚_eventually₁, intermediate_value_univβ‚‚_eventuallyβ‚‚, isConnected_Icc, isConnected_Ici, isConnected_Ico, isConnected_Iic, isConnected_Iio, isConnected_Ioc, isConnected_Ioi, isConnected_Ioo, isConnected_uIoc, isConnected_uIoo, isPreconnected_Icc, isPreconnected_Icc_aux, isPreconnected_Ici, isPreconnected_Ico, isPreconnected_Iic, isPreconnected_Iio, isPreconnected_Ioc, isPreconnected_Ioi, isPreconnected_Ioo, isPreconnected_iff_ordConnected, isPreconnected_uIcc, isPreconnected_uIoc, isPreconnected_uIoo, isTotallyDisconnected_iff_lt, mem_range_of_exists_le_of_exists_ge, ordered_connected_space, setOf_isPreconnected_eq_of_ordered, setOf_isPreconnected_subset_of_ordered
87
Total87

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
strictAnti_of_inj_boundedOrder πŸ“–mathematicalContinuous
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Top.top
OrderTop.toTop
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BoundedOrder.toOrderTop
Bot.bot
OrderBot.toBot
BoundedOrder.toOrderBot
StrictAntiβ€”strictMono_of_inj_boundedOrder
instOrderClosedTopologyOrderDual
strictMonoOn_of_inj_rigidity πŸ“–mathematicalContinuous
Preorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
StrictMonoOn
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.Icc
StrictMonoβ€”min_le_left
le_max_left
LE.le.trans
LE.le.trans'
LT.lt.le
strictMono_of_inj_boundedOrder'
orderTopology_of_ordConnected
Set.ordConnected_Icc
Set.instDenselyOrdered
ContinuousOn.restrict
continuousOn
Set.InjOn.injective
Function.Injective.injOn
strictMono_restrict
Set.strictAntiOn_iff_strictAnti
Set.Icc_subset_Icc
StrictAntiOn.mono
IsAntichain.of_strictMonoOn_antitoneOn
StrictAntiOn.antitoneOn
IsAntichain.not_lt
Set.left_mem_Icc
le_of_lt
Set.right_mem_Icc
min_le_right
le_max_right
StrictMonoOn.mono
strictMono_of_inj πŸ“–mathematicalContinuousStrictMono
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
StrictAnti
β€”strictMonoOn_of_inj_rigidity
instOrderClosedTopologyOrderDual
ContinuousOn.strictMonoOn_of_injOn_Icc'
LT.lt.le
continuousOn
Function.Injective.injOn
subsingleton_or_nontrivial
Subsingleton.strictMono
exists_pair_lt
strictMono_of_inj_boundedOrder πŸ“–mathematicalContinuous
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Bot.bot
OrderBot.toBot
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BoundedOrder.toOrderBot
Top.top
OrderTop.toTop
BoundedOrder.toOrderTop
StrictMonoβ€”lt_of_le_of_ne
LT.lt.ne'
intermediate_value_Ioc
le_top
continuousOn
LT.lt.trans_le
Set.Ioc_top
intermediate_value_Ioo
bot_le
LT.lt.false
LT.lt.trans
lt_of_lt_of_le'
intermediate_value_Ioo'
LT.lt.le
strictMono_of_inj_boundedOrder' πŸ“–mathematicalContinuousStrictMono
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
StrictAnti
β€”strictMono_of_inj_boundedOrder
strictAnti_of_inj_boundedOrder
le_total
surjective πŸ“–β€”Continuous
Filter.Tendsto
Filter.atTop
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.atBot
β€”β€”mem_range_of_exists_le_of_exists_ge
ordered_connected_space
Filter.Eventually.exists
Filter.atBot_neBot
SemilatticeInf.instIsCodirectedOrder
supSet_to_nonempty
Filter.Tendsto.eventually
Filter.eventually_le_atBot
Filter.atTop_neBot
SemilatticeSup.instIsDirectedOrder
Filter.eventually_ge_atTop
surjective' πŸ“–β€”Continuous
Filter.Tendsto
Filter.atBot
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Filter.atTop
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”β€”surjective
instOrderTopologyOrderDual
OrderDual.denselyOrdered

ContinuousOn

Theorems

NameKindAssumesProvesValidatesDepends On
strictAntiOn_of_injOn_Icc πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousOn
Set.Icc
Set.InjOn
StrictAntiOnβ€”strictMonoOn_of_injOn_Icc
instOrderClosedTopologyOrderDual
strictMonoOn_of_injOn_Icc πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousOn
Set.Icc
Set.InjOn
StrictMonoOnβ€”StrictMono.of_restrict
Continuous.strictMono_of_inj_boundedOrder
orderTopology_of_ordConnected
Set.ordConnected_Icc
Set.instDenselyOrdered
restrict
Set.InjOn.injective
strictMonoOn_of_injOn_Icc' πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set.InjOn
StrictMonoOn
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
StrictAntiOn
β€”strictMonoOn_of_injOn_Icc
strictAntiOn_of_injOn_Icc
le_total
strictMonoOn_of_injOn_Ioo πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Ioo
Set.InjOn
StrictMonoOn
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
StrictAntiOn
β€”Set.nonempty_Ioo_subtype
Set.ordConnected_Ioo
orderTopology_of_ordConnected
Set.instDenselyOrdered
restrict
Set.InjOn.injective
strictMono_restrict
Set.strictAntiOn_iff_strictAnti
surjOn_Icc πŸ“–mathematicalContinuousOn
Set
Set.instMembership
Set.SurjOn
Set.Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsPreconnected.intermediate_value
Set.OrdConnected.isPreconnected
surjOn_of_tendsto πŸ“–mathematicalSet.Nonempty
ContinuousOn
Filter.Tendsto
Set.Elem
Set
Set.instMembership
Filter.atBot
Subtype.preorder
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.atTop
Set.SurjOn
Set.univ
β€”Set.surjOn_iff_surjective
Continuous.surjective
Set.Nonempty.to_subtype
orderTopology_of_ordConnected
Set.instDenselyOrdered
restrict
surjOn_of_tendsto' πŸ“–mathematicalSet.Nonempty
ContinuousOn
Filter.Tendsto
Set.Elem
Set
Set.instMembership
Filter.atBot
Subtype.preorder
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Filter.atTop
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.SurjOn
Set.univ
β€”surjOn_of_tendsto
instOrderClosedTopologyOrderDual
surjOn_uIcc πŸ“–mathematicalContinuousOn
Set
Set.instMembership
Set.SurjOn
Set.uIcc
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”le_total
Set.uIcc_of_le
surjOn_Icc
Set.uIcc_of_ge

IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_subset_of_forall_exists_gt πŸ“–mathematicalSet
Set.instMembership
Set.Nonempty
Set.instInter
Set.Ioc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.instHasSubset
Set.Icc
β€”inter
isClosed_Icc
OrderTopology.to_orderClosedTopology
mem_of_ge_of_forall_exists_gt
Set.Ico_subset_Ico_right
Icc_subset_of_forall_exists_lt πŸ“–mathematicalSet
Set.instMembership
Set.Nonempty
Set.instInter
Set.Ico
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.instHasSubset
Set.Icc
β€”inter
isClosed_Icc
OrderTopology.to_orderClosedTopology
mem_of_ge_of_forall_exists_lt
Set.Ioc_subset_Ioc_left
Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset πŸ“–mathematicalSet
Set.instMembership
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.instHasSubset
Set.Icc
β€”lt_or_ge
Set.Icc_eq_empty
Set.left_mem_Icc
Set.Icc_self
le_csSup
csSup_le
LE.le.eq_or_lt
closure_subset_iff
le_rfl
HasSubset.Subset.trans
Set.instIsTransSubset
Set.inter_subset_left
csSup_mem_closure
lt_csSup_iff
LT.lt.le
eq_of_le_of_not_lt
mem_nhdsGT_iff_exists_Ioo_subset'
exists_between
lt_min
ne_of_lt
le_antisymm
le_trans
le_of_lt
le_refl
LE.le.trans
min_le_right
le_or_gt
LE.le.trans_lt
LT.lt.trans_le
min_le_left
Icc_subset_of_forall_mem_nhdsWithin πŸ“–mathematicalSet
Set.instMembership
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.instHasSubset
Set.Icc
β€”Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset
le_rfl
mem_of_ge_of_forall_exists_gt πŸ“–β€”Set
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.Nonempty
Set.instInter
Set.Ioc
β€”β€”Set.left_mem_Icc
csSup_mem
csSup_le
eq_or_lt_of_le
not_lt_of_ge
le_csSup
le_trans
le_of_lt
mem_of_ge_of_forall_exists_lt πŸ“–β€”Set
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.Nonempty
Set.instInter
Set.Ico
β€”β€”mem_of_ge_of_forall_exists_gt
instOrderTopologyOrderDual
Set.Icc_toDual
Set.preimage_inter
Set.mem_image
Equiv.image_symm_eq_preimage
Set.Ico_toDual
Set.Ioc_toDual

IsConnected

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_subset πŸ“–mathematicalIsConnected
Set
Set.instMembership
Set.instHasSubset
Set.Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsPreconnected.Icc_subset
Ioo_csInf_csSup_subset πŸ“–mathematicalIsConnected
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BddAbove
Set
Set.instHasSubset
Set.Ioo
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
β€”isGLB_lt_iff
isGLB_csInf
nonempty
lt_isLUB_iff
isLUB_csSup
Icc_subset
OrderTopology.to_orderClosedTopology
LT.lt.le

IsPreconnected

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_subset πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
Set.instHasSubset
Set.Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”Set.image_id
intermediate_value
continuousOn_id
Iio_csSup_subset πŸ“–mathematicalIsPreconnected
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BddAbove
Set
Set.instHasSubset
Set.Iio
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
β€”Ioi_csInf_subset
instOrderTopologyOrderDual
Ioi_csInf_subset πŸ“–mathematicalIsPreconnected
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BddAbove
Set
Set.instHasSubset
Set.Ioi
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
β€”nonempty_of_not_bddAbove
supSet_to_nonempty
isGLB_lt_iff
isGLB_csInf
not_bddAbove_iff
Icc_subset
OrderTopology.to_orderClosedTopology
LT.lt.le
eq_univ_of_unbounded πŸ“–mathematicalIsPreconnected
BddBelow
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
BddAbove
Set.univβ€”Set.eq_univ_of_forall
not_bddBelow_iff
not_bddAbove_iff
Icc_subset
le_of_lt
intermediate_value πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
ContinuousOn
Set.instHasSubset
Set.Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”intermediate_valueβ‚‚
continuousOn_const
intermediate_value_Ici πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
Filter.atTop
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.instHasSubset
Set.Ici
Set.image
β€”intermediate_valueβ‚‚_eventually₁
continuousOn_const
Filter.tendsto_atTop
intermediate_value_Ico πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
nhds
Set.instHasSubset
Set.Ico
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”intermediate_valueβ‚‚_eventually₁
continuousOn_const
Filter.Tendsto.eventually_const_le
instClosedIicTopology
intermediate_value_Iic πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
Filter.atBot
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.instHasSubset
Set.Iic
Set.image
β€”intermediate_valueβ‚‚_eventually₁
continuousOn_const
Filter.tendsto_atBot
intermediate_value_Iii πŸ“–mathematicalIsPreconnected
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
Filter.atBot
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.atTop
Set
Set.instHasSubset
Set.univ
Set.image
β€”intermediate_valueβ‚‚_eventuallyβ‚‚
continuousOn_const
Filter.Tendsto.eventually_le_atBot
Filter.Tendsto.eventually_ge_atTop
intermediate_value_Iio πŸ“–mathematicalIsPreconnected
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
Filter.atBot
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhds
Set
Set.instHasSubset
Set.Iio
Set.image
β€”intermediate_valueβ‚‚_eventuallyβ‚‚
continuousOn_const
Filter.Tendsto.eventually_le_atBot
Filter.Tendsto.eventually_const_le
instClosedIicTopology
intermediate_value_Ioc πŸ“–mathematicalIsPreconnected
Set
Set.instMembership
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
nhds
Set.instHasSubset
Set.Ioc
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”intermediate_valueβ‚‚_eventually₁
continuousOn_const
Filter.Tendsto.eventually_le_const
instClosedIciTopology
intermediate_value_Ioi πŸ“–mathematicalIsPreconnected
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
nhds
Filter.atTop
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instHasSubset
Set.Ioi
Set.image
β€”intermediate_valueβ‚‚_eventuallyβ‚‚
continuousOn_const
Filter.Tendsto.eventually_le_const
instClosedIciTopology
Filter.Tendsto.eventually_ge_atTop
intermediate_value_Ioo πŸ“–mathematicalIsPreconnected
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.Tendsto
nhds
Set
Set.instHasSubset
Set.Ioo
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”intermediate_valueβ‚‚_eventuallyβ‚‚
continuousOn_const
Filter.Tendsto.eventually_le_const
instClosedIciTopology
Filter.Tendsto.eventually_const_le
instClosedIicTopology
intermediate_valueβ‚‚ πŸ“–β€”IsPreconnected
Set
Set.instMembership
ContinuousOn
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”β€”intermediate_value_univβ‚‚
Subtype.preconnectedSpace
continuousOn_iff_continuous_restrict
intermediate_valueβ‚‚_eventually₁ πŸ“–β€”IsPreconnected
Set
Set.instMembership
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.EventuallyLE
β€”β€”intermediate_value_univβ‚‚_eventually₁
Subtype.preconnectedSpace
Filter.comap_coe_neBot_of_le_principal
continuousOn_iff_continuous_restrict
Filter.Eventually.comap
Subtype.prop
intermediate_valueβ‚‚_eventuallyβ‚‚ πŸ“–mathematicalIsPreconnected
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
ContinuousOn
Filter.EventuallyLE
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instMembership
β€”intermediate_value_univβ‚‚_eventuallyβ‚‚
Subtype.preconnectedSpace
Filter.comap_coe_neBot_of_le_principal
continuousOn_iff_continuous_restrict
Filter.Eventually.comap
Subtype.prop
mem_intervals πŸ“–mathematicalIsPreconnectedSet
Set.instMembership
Set.instInsert
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
Set.Ico
Set.Ioc
Set.Ioo
Set.Ici
Set.Ioi
Set.Iic
Set.Iio
Set.univ
Set.instSingletonSet
Set.instEmptyCollection
β€”Set.eq_empty_or_nonempty
Set.mem_singleton
Set.mem_of_subset_of_mem
Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
IsConnected.Ioo_csInf_csSup_subset
subset_Icc_csInf_csSup
Set.mem_Ici_Ioi_of_subset_of_subset
Ioi_csInf_subset
csInf_le
Set.mem_Iic_Iio_of_subset_of_subset
Iio_csSup_subset
le_csSup
eq_univ_of_unbounded
OrderTopology.to_orderClosedTopology
ordConnected πŸ“–mathematicalIsPreconnectedSet.OrdConnected
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”Icc_subset

Set.OrdConnected

Theorems

NameKindAssumesProvesValidatesDepends On
isPreconnected πŸ“–mathematicalβ€”IsPreconnectedβ€”isPreconnected_of_forall_pair
uIcc_subset
Set.left_mem_uIcc
Set.right_mem_uIcc
isPreconnected_uIcc

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
eq_Icc_csInf_csSup_of_connected_bdd_closed πŸ“–mathematicalIsConnected
BddBelow
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
BddAbove
Set.Icc
InfSet.sInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
SupSet.sSup
ConditionallyCompletePartialOrderSup.toSupSet
β€”HasSubset.Subset.antisymm
Set.instAntisymmSubset
subset_Icc_csInf_csSup
IsConnected.Icc_subset
OrderTopology.to_orderClosedTopology
IsClosed.csInf_mem
IsConnected.nonempty
IsClosed.csSup_mem
exists_mem_Icc_isFixedPt πŸ“–mathematicalContinuousOn
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Preorder.toLE
Set
Set.instMembership
Function.IsFixedPt
β€”IsPreconnected.intermediate_valueβ‚‚
OrderTopology.to_orderClosedTopology
isPreconnected_Icc
Set.right_mem_Icc
Set.left_mem_Icc
continuousOn_id
exists_mem_Icc_isFixedPt_of_mapsTo πŸ“–mathematicalContinuousOn
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Preorder.toLE
Set.MapsTo
Set
Set.instMembership
Function.IsFixedPt
β€”exists_mem_Icc_isFixedPt
Set.left_mem_Icc
Set.right_mem_Icc
exists_mem_uIcc_isFixedPt πŸ“–mathematicalContinuousOn
Set.uIcc
ConditionallyCompleteLattice.toLattice
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Preorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
Set
Set.instMembership
Function.IsFixedPt
β€”IsPreconnected.intermediate_valueβ‚‚
OrderTopology.to_orderClosedTopology
isPreconnected_uIcc
Set.right_mem_uIcc
Set.left_mem_uIcc
continuousOn_id
intermediate_value_Icc πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”IsPreconnected.intermediate_value
isPreconnected_Icc
Set.left_mem_Icc
Set.right_mem_Icc
intermediate_value_Icc' πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”IsPreconnected.intermediate_value
isPreconnected_Icc
Set.right_mem_Icc
Set.left_mem_Icc
intermediate_value_Ico πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ico
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”eq_or_lt_of_le
not_lt_of_ge
IsPreconnected.intermediate_value_Ico
isPreconnected_Ico
refl
instReflLe
right_nhdsWithin_Ico_neBot
inf_le_right
ContinuousOn.mono
Set.Ico_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
intermediate_value_Ico' πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ioc
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
Set.Ico
β€”eq_or_lt_of_le
not_lt_of_ge
IsPreconnected.intermediate_value_Ioc
isPreconnected_Ico
refl
instReflLe
right_nhdsWithin_Ico_neBot
inf_le_right
ContinuousOn.mono
Set.Ico_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
intermediate_value_Ioc πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ioc
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”eq_or_lt_of_le
not_le_of_gt
IsPreconnected.intermediate_value_Ioc
isPreconnected_Ioc
refl
instReflLe
left_nhdsWithin_Ioc_neBot
inf_le_right
ContinuousOn.mono
Set.Ioc_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
intermediate_value_Ioc' πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ico
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
Set.Ioc
β€”eq_or_lt_of_le
not_le_of_gt
IsPreconnected.intermediate_value_Ico
isPreconnected_Ioc
refl
instReflLe
left_nhdsWithin_Ioc_neBot
inf_le_right
ContinuousOn.mono
Set.Ioc_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
intermediate_value_Ioo πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ioo
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”eq_or_lt_of_le
not_lt_of_gt
IsPreconnected.intermediate_value_Ioo
isPreconnected_Ioo
left_nhdsWithin_Ioo_neBot
right_nhdsWithin_Ioo_neBot
inf_le_right
ContinuousOn.mono
Set.Ioo_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
refl
instReflLe
intermediate_value_Ioo' πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
ContinuousOn
Set.Icc
Set
Set.instHasSubset
Set.Ioo
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”eq_or_lt_of_le
not_lt_of_gt
IsPreconnected.intermediate_value_Ioo
isPreconnected_Ioo
right_nhdsWithin_Ioo_neBot
left_nhdsWithin_Ioo_neBot
inf_le_right
ContinuousOn.mono
Set.Ioo_subset_Icc_self
ContinuousWithinAt.mono
ContinuousOn.continuousWithinAt
refl
instReflLe
intermediate_value_uIcc πŸ“–mathematicalContinuousOn
Set.uIcc
ConditionallyCompleteLattice.toLattice
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set
Set.instHasSubset
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.image
β€”le_total
Set.uIcc_of_le
IsPreconnected.intermediate_value
isPreconnected_uIcc
Set.uIcc_of_ge
intermediate_value_univ πŸ“–mathematicalContinuousSet
Set.instHasSubset
Set.Icc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.range
β€”intermediate_value_univβ‚‚
continuous_const
intermediate_value_univβ‚‚ πŸ“–β€”Continuous
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”β€”isPreconnected_closed_iff
PreconnectedSpace.isPreconnected_univ
isClosed_le
le_total
le_antisymm
intermediate_value_univβ‚‚_eventually₁ πŸ“–β€”Continuous
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.EventuallyLE
β€”β€”Filter.Eventually.exists
intermediate_value_univβ‚‚
intermediate_value_univβ‚‚_eventuallyβ‚‚ πŸ“–β€”Continuous
Filter.EventuallyLE
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”β€”Filter.Eventually.exists
intermediate_value_univβ‚‚
isConnected_Icc πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
IsConnected
Set.Icc
β€”Set.nonempty_Icc
isPreconnected_Icc
isConnected_Ici πŸ“–mathematicalβ€”IsConnected
Set.Ici
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.nonempty_Ici
isPreconnected_Ici
isConnected_Ico πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
IsConnected
Set.Ico
β€”Set.nonempty_Ico
isPreconnected_Ico
isConnected_Iic πŸ“–mathematicalβ€”IsConnected
Set.Iic
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.nonempty_Iic
isPreconnected_Iic
isConnected_Iio πŸ“–mathematicalβ€”IsConnected
Set.Iio
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.nonempty_Iio
isPreconnected_Iio
isConnected_Ioc πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
IsConnected
Set.Ioc
β€”Set.nonempty_Ioc
isPreconnected_Ioc
isConnected_Ioi πŸ“–mathematicalβ€”IsConnected
Set.Ioi
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.nonempty_Ioi
isPreconnected_Ioi
isConnected_Ioo πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
IsConnected
Set.Ioo
β€”Set.nonempty_Ioo
isPreconnected_Ioo
isConnected_uIoc πŸ“–mathematicalβ€”IsConnected
Set.uIoc
ConditionallyCompleteLinearOrder.toLinearOrder
β€”Set.nonempty_uIoc
isPreconnected_uIoc
isConnected_uIoo πŸ“–mathematicalβ€”IsConnected
Set.uIoo
ConditionallyCompleteLinearOrder.toLinearOrder
β€”Set.nonempty_uIoo
isPreconnected_uIoo
isPreconnected_Icc πŸ“–mathematicalβ€”IsPreconnected
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”isPreconnected_closed_iff
le_total
isPreconnected_Icc_aux
Set.inter_comm
Set.union_comm
isPreconnected_Icc_aux πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set
Set.instHasSubset
Set.Icc
Set.instUnion
Set.instMembership
Set.instInter
Set.Nonemptyβ€”Set.Icc_subset_Icc
IsClosed.Icc_subset_of_forall_mem_nhdsWithin
IsClosed.inter
isClosed_Icc
OrderTopology.to_orderClosedTopology
Set.Ico_subset_Icc_self
nhdsWithin_Ioc_eq_nhdsGT
instClosedIciTopology
mem_nhdsWithin
IsClosed.isOpen_compl
Set.Subset.rfl
Filter.mem_of_superset
le_trans
le_of_lt
Set.right_mem_Icc
isPreconnected_Ici πŸ“–mathematicalβ€”IsPreconnected
Set.Ici
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Ici
isPreconnected_Ico πŸ“–mathematicalβ€”IsPreconnected
Set.Ico
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Ico
isPreconnected_Iic πŸ“–mathematicalβ€”IsPreconnected
Set.Iic
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Iic
isPreconnected_Iio πŸ“–mathematicalβ€”IsPreconnected
Set.Iio
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Iio
isPreconnected_Ioc πŸ“–mathematicalβ€”IsPreconnected
Set.Ioc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Ioc
isPreconnected_Ioi πŸ“–mathematicalβ€”IsPreconnected
Set.Ioi
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Ioi
isPreconnected_Ioo πŸ“–mathematicalβ€”IsPreconnected
Set.Ioo
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.OrdConnected.isPreconnected
Set.ordConnected_Ioo
isPreconnected_iff_ordConnected πŸ“–mathematicalβ€”IsPreconnected
Set.OrdConnected
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”IsPreconnected.ordConnected
OrderTopology.to_orderClosedTopology
Set.OrdConnected.isPreconnected
isPreconnected_uIcc πŸ“–mathematicalβ€”IsPreconnected
Set.uIcc
ConditionallyCompleteLattice.toLattice
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”isPreconnected_Icc
isPreconnected_uIoc πŸ“–mathematicalβ€”IsPreconnected
Set.uIoc
ConditionallyCompleteLinearOrder.toLinearOrder
β€”isPreconnected_Ioc
isPreconnected_uIoo πŸ“–mathematicalβ€”IsPreconnected
Set.uIoo
ConditionallyCompleteLinearOrder.toLinearOrder
β€”isPreconnected_Ioo
isTotallyDisconnected_iff_lt πŸ“–mathematicalβ€”IsTotallyDisconnected
Set
Set.instMembership
Set.Ioo
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
β€”Set.not_nontrivial_iff
Set.not_ordConnected_inter_Icc_iff
Set.inter_subset_left
le_rfl
LT.lt.le
Set.OrdConnected.out'
mem_range_of_exists_le_of_exists_ge πŸ“–mathematicalContinuous
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instMembership
Set.range
β€”intermediate_value_univ
ordered_connected_space πŸ“–mathematicalβ€”PreconnectedSpaceβ€”Set.OrdConnected.isPreconnected
Set.ordConnected_univ
setOf_isPreconnected_eq_of_ordered πŸ“–mathematicalβ€”setOf
Set
IsPreconnected
Set.instUnion
Set.range
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.Ico
Set.Ioc
Set.Ioo
Set.Ici
Set.Ioi
Set.Iic
Set.Iio
Set.instInsert
Set.univ
Set.instSingletonSet
Set.instEmptyCollection
β€”Set.Subset.antisymm
setOf_isPreconnected_subset_of_ordered
ordered_connected_space
setOf_isPreconnected_subset_of_ordered πŸ“–mathematicalβ€”Set
Set.instHasSubset
setOf
IsPreconnected
Set.instUnion
Set.range
Set.Icc
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Set.Ico
Set.Ioc
Set.Ioo
Set.Ici
Set.Ioi
Set.Iic
Set.Iio
Set.instInsert
Set.univ
Set.instSingletonSet
Set.instEmptyCollection
β€”IsPreconnected.mem_intervals
Set.union_insert
Set.union_singleton

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