Lemma I.10.10 in Kunen's set thoery (2011) claims that a set $A$ is well-orderable iff $A$ is bijective with an ordinal.
He does not prove this, and one direction is clear to me.
However, I can't see why $A$ bijective with an ordinal implies $A$ is well-orderable. By the way he develops this within ZFC except foundation, power set, and choice.
Any help is appreciated -Thanks!