Hausdorff's Maximal Principle implies Axiom of Choice/Lemma

Theorem
Let $S$ be a non-empty set.

Let $\Sigma$ be the set of all choice functions of all non-empty subsets of $S$.

$\Sigma$ is closed under chain unions.

Proof
Let $f$ and $g$ be choice functions.

Then:
 * $f \subseteq g$


 * $\Dom f \subseteq \Dom g$
 * $\forall x \in \Dom f: \map f x = \map g x$
 * $\forall x \in \Dom f: \map f x = \map g x$