Page "Well-ordering theorem" Paragraph 3

The well ordering theorem follows easily from Zorn's Lemma.

Take the set A of all well orderings of subsets of X: an element of A is an ordered pair ( a, b ) where a is a subset of X and b is a well ordering of a.

A can be partially ordered by continuation.

That means, define E ≤ F if E is an initial segment of F and the ordering of the members in E is the same as their ordering in F. If E is a chain in A, then the union of the sets in E can be ordered in a way that makes it a continuation of any set in E ; this ordering is a well ordering, and therefore, an upper bound of E in A.

We may therefore apply Zorn's Lemma to conclude that A has a maximal element, say ( M, R ).

The set M must be equal to X, for if X has an element x not in M, then the set M ∪