Well-ordering theorem

Well-ordering theorem: For every set $X$, there exists a total order $\le$ on $X$ such that $(X,\le)$ is a well-ordered set .

In the presence of the other axioms of set theory (ZF), the well-ordering theorem is equivalent to the axiom of choice (and also to Zorn's lemma ).

Proof sketch (idea): Using choice (or an equivalent maximality principle), one constructs a well-order by successively selecting “next” elements from the remaining set; well-foundedness comes from the way the selection process is set up.