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.

That is, every set is well-orderable.

Sufficient Condition
Let it be supposed that every set is well-orderable.

Then by the converse to Zermelo's Well-Ordering Theorem:
 * the Axiom of Choice holds.