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) c → Set.Nonempty c → ⊤ ∉ c → ∃ (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) c → Set.Nonempty c → ⊥ ∉ c → ∃ (x : α) (_ : x ≠ ⊥), x ∈ lowerBounds c)
:
IsAtomic α
Zorn's lemma: A partial order is atomic if every nonempty chain c, ⊥ ∉ c, has a lower
bound not equal to ⊥.