Axiom of Choice implies Kuratowski's Lemma

Theorem
Let the Axiom of Choice be accepted.

Then Kuratowski's Lemma holds.

Proof
Kuratowski's Lemma states that:

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.