Axiom of Choice implies Kuratowski's Lemma/Proof 2

Proof
Recall Kuratowski's Lemma:

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.