Ordinal Addition is Closed

Theorem
Let $\operatorname{On}$ be the class of all ordinals.

Then:
 * $\forall x, y \in \operatorname{On}: \left({x + y}\right) \in \operatorname{On}$

That is: the sum $x+y$ is an ordinal.

Proof
Using Transfinite Induction on $y$: