Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice

Theorem
Zermelo's Well-Ordering Theorem holds the Axiom of Choice holds.

That is, every set is well-orderable every collection of sets has a choice function.

Necessary Condition
Suppose the Axiom of Choice holds.

Then Zermelo's Well-Ordering Theorem holds by Zermelo's Well-Ordering Theorem itself.

Sufficient Condition
Assume Zermelo's Well-Ordering Theorem holds.

Let $\FF$ be an arbitrary collection of sets.

By assumption all sets can be well-ordered.

Hence the set $\bigcup \FF$ of all elements of sets contained in $\FF$ is well-ordered by some ordering $<$.

By definition, in a well-ordered set, every subset has a unique least element.

Also, note that each set in $\FF$ is a subset of $\bigcup \FF$.

Thus, we may define the choice function $c$:
 * $\forall X \in \FF: c: \FF \to \bigcup \FF$

by letting $\map c X$ be the least element of $X$ under $<$.