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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} [PartialOrder α] [OrderTop α] (h : ∀ (c : Set α), IsChain (fun (x x_1 : α) => x x_1) cSet.Nonempty cc∃ (x : α) (_ : x ), x upperBounds c) :

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} [PartialOrder α] [OrderBot α] (h : ∀ (c : Set α), IsChain (fun (x x_1 : α) => x x_1) cSet.Nonempty cc∃ (x : α) (_ : x ), x lowerBounds c) :

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