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Theorems · Theorem · order theory

bot_notMem_iff

∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {s : Set α}, ⊥ ∉ s ↔ ∀ x ∈ s, ⊥ < x
Defined in
Mathlib.Order.BoundedOrder.Basic
Cited by
2 results in Mathlib
Foundations
Depth 10 from the axioms · uses no axioms
Assumes
PartialOrderOrderBot
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

zero_notMem_iff · cited by 0zero_notMem_iffone_notMem_iff · cited by 0one_notMem_iffSet · cited by 53352SetPartialOrder · cited by 6410PartialOrderBot.bot · cited by 4720Bot.botOrderBot · cited by 1055OrderBotNe.bot_lt · cited by 116Ne.bot_ltLT.lt.false · cited by 66lt.falsebot_notMem_iffCITED BYCITES

Cites6

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.