Class of All Ordinals is Ordinal

Theorem
The Ordinal Class $\operatorname{On}$ is an ordinal.

Proof
The membership relation is equivalent to the subset relation when restricted to ordinals by Ordinal Proper Subset Membership.

It follows that:


 * $\forall x \in \operatorname{On}: x \subset \operatorname{On}$

The segment of the class of ordinals is:


 * $\left\{ x \in \operatorname{On} : x \subset \operatorname{On} \right\}$

This class is equal to $\operatorname{On}$.

Therefore, by the definition of ordinal, $\operatorname{On}$ is an ordinal.

Source

 * :$7.12$