Axiom of Choice implies Tukey's Lemma

Theorem
Let the Axiom of Choice be accepted.

Then Tukey's Lemma holds.

Proof
Let us assume the truth of the Axiom of Choice.

From Axiom of Choice implies Kuratowski's Lemma, it follows that Kuratowski's Lemma holds.

From Kuratowski's Lemma implies Tukey's Lemma, it follows that Tukey's Lemma holds.

Hence the result.