Isomorphic Ordinals are Equal

Theorem
Let $S, T$ be ordinals.

Let $S \cong T$, that is, let $X$ and $Y$ be order isomorphic.

Then $S = T$.

Proof
Let $S \cong T$.

Suppose that $S \ne T$.

Then from Relation between Unequal Ordinals, either $S$ is a segment of $T$ or $T$ is a segment of $S$.

But as $S \cong T$, from No Isomorphism from Woset to Segment, neither $S$ nor $T$ can be a segment of the other.

From this contradiction it follows that $S = T$.