Distinct Ordinals are not Order Isomorphic

Theorem
Let $\alpha$ and $\beta$ be ordinals such that $\alpha \ne \beta$.

Then $\alpha$ and $\beta$ are not order isomorphic.

Proof
By definition, an ordinal is well-ordered by the subset relation.

From Class of All Ordinals is Well-Ordered by Subset Relation, the class of all ordinals is a nest.

Hence:
 * $\paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha}$

The result follows by Well-Ordered Class is not Isomorphic to Initial Segment.