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Theorems · Theorem · order theory

subsingleton_iff_bot_eq_top

∀ {α : Type u} [inst : PartialOrder α] [inst_1 : BoundedOrder α], ⊥ = ⊤ ↔ Subsingleton α
Defined in
Mathlib.Order.BoundedOrder.Basic
Cited by
16 results in Mathlib
Foundations
Depth 7 from the axioms · uses no axioms
Assumes
PartialOrderBoundedOrder
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Submodule.subsingleton_iff · cited by 7Submodule.subsingleton_iffIsSemisimpleModule.eq_bot_or_exists_simple_le · cited by 4IsSemisimpleModule.eq_bot…IsLocalRing.subsingleton_tensorProduct · cited by 4IsLocalRing.subsingleton_…LieSubmodule.subsingleton_iff · cited by 4LieSubmodule.subsingleton…AddGroup.nilpotencyClass_zero_iff_subsingleton · cited by 2AddGroup.nilpotencyClass_…Group.nilpotencyClass_zero_iff_subsingleton · cited by 2Group.nilpotencyClass_zer…IsCyclotomicExtension.iff_union_singleton_one · cited by 2IsCyclotomicExtension.iff…AddGroup.rank_eq_zero_iff · cited by 2AddGroup.rank_eq_zero_iffIsCyclotomicExtension.finite · cited by 1IsCyclotomicExtension.fin…Fin.zero_eq_top · cited by 1Fin.zero_eq_topConvexCone.isGenerating_bot_iff · cited by 1ConvexCone.isGenerating_b…LieModule.nilpotencyLength_eq_zero_iff · cited by 1LieModule.nilpotencyLengt…RingTheory.Sequence.IsRegular.nil · cited by 0IsRegular.nilIsCyclotomicExtension.isCyclotomicExtension_zero_iff · cited by 0IsCyclotomicExtension.isC…Subfield.bot_eq_of_charZero · cited by 0Subfield.bot_eq_of_charZe…Top.top · cited by 9680Top.topPartialOrder · cited by 6410PartialOrderBot.bot · cited by 4720Bot.botBoundedOrder · cited by 270BoundedOrdersubsingleton_of_bot_eq_top · cited by 8subsingleton_of_bot_eq_topsubsingleton_iff_bot_eq_topCITED BYCITES

Cites5

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by16

Results whose statement or proof uses this declaration.