Tukey's Lemma implies Zorn's Lemma

Theorem
Let Tukey's Lemma be accepted as true.

Then Zorn's Lemma holds.

Proof
Recall Tukey's Lemma:

Recall Zorn's Lemma:

So, let us assume Tukey's Lemma.

Let $S$ be a non-empty ordered set, with $T$ as defined.

From Property of being Totally Ordered is of Finite Character:
 * $T$ is of finite character.

From Ordering on Singleton is Total Ordering it follows trivially that $T$ is non-empty.

Then by Tukey's Lemma:
 * every element of $T$ is a subset of a maximal element of $T$ under the subset relation.

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