Counting Theorem/Corollary

Corollary to Counting Theorem
Every properly well-ordered proper class is order isomorphic to the class of all ordinals.

Proof
Let $A$ be a properly well-ordered class.

Let $\On$ denote the class of all ordinals.

By the Axiom of Replacement, neither $A$ nor $\On$ can be order isomorphic to a proper lower section of the other.

Hence it must be that $A$ is order isomorphic to $\On$.