Axiom of Choice implies Kuratowski's Lemma/Proof 2

Theorem

Let the Axiom of Choice be accepted.

Then Kuratowski's Lemma holds.

Proof

Recall Kuratowski's Lemma:

Let $S$ be a set of sets which is closed under chain unions.

Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

By the Axiom of Choice, there exists a choice function for $S$.

By Closed Set under Chain Unions with Choice Function is of Type $M$:

every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Hence the result.

$\blacksquare$

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 $5.2$