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Theorems · Theorem · order theory

IsCoatom.le_iff

∀ {α : Type u_2} [inst : PartialOrder α] [inst_1 : OrderTop α] {a x : α}, IsCoatom a → (a ≤ x ↔ x = ⊤ ∨ x = a)
Defined in
Mathlib.Order.Atoms
Cited by
4 results in Mathlib
Foundations
Depth 16 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
PartialOrderOrderTop
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Around this declaration

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

GaloisInsertion.isCoatom_iff · cited by 2GaloisInsertion.isCoatom_…IsCoatom.codisjoint_of_ne · cited by 2IsCoatom.codisjoint_of_neSubmodule.isClosed_or_dense_of_isCoatom · cited by 1Submodule.isClosed_or_den…IsCoatom.ne_iff_eq_top · cited by 1IsCoatom.ne_iff_eq_topTop.top · cited by 9680Top.topPartialOrder · cited by 6410PartialOrderOrderTop · cited by 493OrderTopIsCoatom · cited by 114IsCoatomIsAtom.le_iff · cited by 10IsAtom.le_iffIsCoatom.dual · cited by 6IsCoatom.dualIsCoatom.le_iffCITED BYCITES

Cites6

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

Cited by4

Results whose statement or proof uses this declaration.