Class of All Ordinals is Ordinal

Theorem
The class of all ordinals $\On$ 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:


 * $\forall x \in \On: x \subset \On$

The initial segment of the class of all ordinals is:


 * $\set {x \in \On : x \subset \On}$

This class is equal to $\On$.

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