Theorems · Theorem · order theory

ChainCompletePartialOrder.subset_bot_iff

∀ {α : Type u_1} [inst : ChainCompletePartialOrder α] {x : α} {f : α → α} {s : Set α},
  ChainCompletePartialOrder.IsAdmissible x f s →
    (s ⊆ ChainCompletePartialOrder.bot x f ↔ s = ChainCompletePartialOrder.bot x f)

Defined in: Mathlib.Order.BourbakiWitt
Cited by: 2 results in Mathlib
Foundations: Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: ChainCompletePartialOrder

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Cites7

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

Setstatement and proof · cited by 53,352
subset_antisymmproof · cited by 150
Set.sInter_subset_of_memproof · cited by 28
ChainCompletePartialOrderstatement and proof · cited by 19
ChainCompletePartialOrder.botstatement and proof · cited by 12
subset_reflproof · cited by 11
ChainCompletePartialOrder.IsAdmissiblestatement and proof · cited by 8

Cited by2

Results whose statement or proof uses this declaration.

ChainCompletePartialOrder.IsExtremePt.bot_eq_of_le_or_map_leproof · cited by 2
ChainCompletePartialOrder.IsExtremePt.setOfPred_isExtremePt_eq_botproof · cited by 2