Well-Ordering Theorem is Equivalent to the 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 ($\Longleftarrow$) 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 $<$.

$\blacksquare$