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.