Ordinal is Subset of Class of All Ordinals

Theorem
Suppose $A$ is an ordinal. Then, $A \subseteq \operatorname{On}$ where $\operatorname{On}$ represents the class of all ordinals.

Proof
By Ordinal Member of Ordinal Class, $A \in \operatorname{On} \lor A = \operatorname{On}$. In either case, $A \subseteq \operatorname{On}$ since $\operatorname{On}$ is transitive.

Source

 * :$7.15$