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:

$f \subseteq g$

if and only if:

$\Dom f \subseteq \Dom g$

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

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles: Theorem $5.12$