Class of All Ordinals is Ordinal

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

Proof
The epsilon relation is equivalent to the strict subset relation when restricted to ordinals by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

It follows that:


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

The initial 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

 * : $7.12$