[Trichotomy of ordinals]

Let be ordinals. Then, it satisfies one of the alternatives, .

Lemma 1. Let be an ordinal. Then, any member of is itself an ordinal number.

If , we are done. It is an ordinal number. Otherwise, it is a member of some ordinal number in X. By lemma 1, it is an ordinal number.

We show that is the least element of X. If , we are done.

If , we claim that x is the least element (w.r.t -image) in X.

Take any . If , we are done again. Otherwise, if , then or y=x by trichotomy of ordinals. This implies that . Since y is an element of X, . This forces y=x. Thus x is the least element of X.