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

Theorem
The Well-Ordering Theorem holds the Axiom of Choice holds.

That is, every set can be well-ordered every collection of sets has a choice function.

Necessary Condition
Suppose the Axiom of Choice holds.

Then the Well-Ordering Theorem holds by the Well-Ordering Theorem itself.

Sufficient Condition
Assume the Well-Ordering Theorem holds.

Let $\mathcal F$ be an arbitrary collection of sets.

By assumption all sets can be well-ordered.

Hence the set $\bigcup \mathcal F$ of all elements of sets contained in $\mathcal F$ 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 $\mathcal F$ is a subset of $\bigcup \mathcal F$.

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

by letting $c \left({X}\right)$ be the least element of $X$ under $<$.