# Documentation

Mathlib.Order.ZornAtoms

# Zorn lemma for (co)atoms #

In this file we use Zorn's lemma to prove that a partial order is atomic if every nonempty chain c, ⊥ ∉ c, has a lower bound not equal to ⊥. We also prove the order dual version of this statement.

theorem IsCoatomic.of_isChain_bounded {α : Type u_1} [inst : ] [inst : ] (h : ∀ (c : Set α), IsChain (fun x x_1 => x x_1) c¬ cx x_1, x ) :

Zorn's lemma: A partial order is coatomic if every nonempty chain c, ⊤ ∉ c, has an upper bound not equal to ⊤.

theorem IsAtomic.of_isChain_bounded {α : Type u_1} [inst : ] [inst : ] (h : ∀ (c : Set α), IsChain (fun x x_1 => x x_1) c¬ cx x_1, x ) :

Zorn's lemma: A partial order is atomic if every nonempty chain c, ⊥ ∉ c, has an lower bound not equal to ⊥.