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Theorems · Theorem · order theory

IsLowerSet.eq_empty_or_Iic

∀ {α : Type u_1} [inst : LinearOrder α] {s : Set α} [WellFoundedGT α], IsLowerSet s → s = ∅ ∨ ∃ a, s = Set.Iic a
Defined in
Mathlib.Order.UpperLower.Basic
Cited by
2 results in Mathlib
Foundations
Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
LinearOrderWellFoundedGT
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

IsLowerSet.isClosed · cited by 1IsLowerSet.isClosedIsUpperSet.eq_univ_or_Ioi · cited by 0IsUpperSet.eq_univ_or_IoiSet · cited by 53352SetLinearOrder · cited by 8572LinearOrderSet.ext · cited by 2266Set.extSet.Iic · cited by 1111Set.IicIsLowerSet · cited by 167IsLowerSetWellFoundedGT · cited by 114WellFoundedGTSet.nonempty_iff_ne_empty · cited by 96Set.nonempty_iff_ne_emptyWellFounded.min · cited by 33WellFounded.minWellFounded.min_mem · cited by 23WellFounded.min_memwellFounded_gt · cited by 5wellFounded_gtIsLowerSet.eq_empty_or_IicCITED BYCITES

Cites10

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

Cited by2

Results whose statement or proof uses this declaration.