Isomorphic Ordinals are Equal
Then $A = B$.
Let $S \cong T$.
Suppose that $S \ne T$.
Then from the corollary to Relation between Two Ordinals, either:
- $S$ is an initial segment of $T$
- $T$ is an initial segment of $S$.
From this contradiction it follows that $S = T$.
Hence, neither $A$ nor $B$ can be an element of the other.
By Ordinal Membership is Trichotomy, it follows that $A = B$.