Class of All Ordinals is Only Proper Class of Ordinals

Theorem
Let $A$ be a transitive proper class of ordinals.

Then $A$ is the class of all ordinals $\On$.

Proof
Let $A$ be a transitive class of ordinals.

Let there exist $\alpha \in \On$ such that $\alpha \notin A$.

Then by Transitive Class of Ordinals is Subset of Ordinal not in it:
 * $A \subseteq \alpha$

But that makes $A$ a set.

So if $A$ is a proper class, it must contain all ordinals.

That is:
 * $A = \On$