Theorems · Theorem · group theory

Submonoid.mem_iSup_of_directed

∀ {M : Type u_1} [inst : MulOneClass M] {ι : Sort u_4} [Nonempty ι] {S : ι → Submonoid M},
  Directed (fun x1 x2 => x1 ≤ x2) S → ∀ {x : M}, x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i

Defined in

Mathlib.Algebra.Group.Submonoid.Membership

Cited by

3 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MulOneClassNonempty