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 $$S = T$$.