Axiom of Choice implies Zorn's Lemma

Theorem
Acceptance of the Axiom of Choice implies the truth of Zorn's Lemma.

Statement of Zorn's Lemma
Let $\left({X, \preceq}\right), X \ne \varnothing$ be a non-empty ordered set such that every non-empty chain in $X$ has an upper bound in $X$.

Then $X$ has at least one maximal element.