Zorn's Lemma

Let be a (nonempty) Poset. Suppose every Chain in has an Upper Bound. Then has a Maximal element.

Proof

Assume has no maximal element. For each fix s.t. (by Axiom of Choice) Also, for each chain , let be an upper bound for (by Axiom of Choice) Let (Hartogs’ Lemma) Define by recursion:

An easy induction (on with fixed) shows that Hence is injective which is a contradiction by Hartogs’ Lemma.