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

Theorem
The Well-Ordering Theorem holds if and only if the Axiom of Choice holds.

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

Proof
The direction ($\impliedby$) is the proof of the Well-Ordering Theorem.

Assume the Well-Ordering Theorem, and let $\mathcal F$ be an arbitrary collection of sets.

Since by assumption all sets are well-orderable, 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 $c: \mathcal F \to \bigcup \mathcal F$ for each $X \in \mathcal F$ by letting $c(X)$ be the least element of $X$ under $<$.