Isomorphic Ordinals are Equal

Theorem
Let $A$ and $B$ be ordinals that are order isomorphic.

Then $A = B$.

Proof
From No Isomorphism from Woset to Segment, neither $A$ nor $B$ can be an initial segment of the other.

By definition, every element of an ordinal is an initial segment of it; hence, neither $A$ nor $B$ can be an element of the other.

By Ordinal Membership Trichotomy, it follows that $A = B$.