Kuratowski's Lemma implies Tukey's Lemma

Theorem
Let Kuratowski's Lemma be accepted as true.

Then Tukey's Lemma holds.

Proof
Recall Kuratowski's Lemma:

Recall Tukey's Lemma:

So, let us assume Kuratowski's Lemma.

Let $S$ be a non-empty set of finite character.

From Class of Finite Character is Closed under Chain Unions, $S$ is closed under chain unions.

Then by Kuratowski's Lemma:
 * every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Thus it is seen that Tukey's Lemma likewise holds.