Class of All Ordinals is Ordinal

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

Proof
All ordinals are not elements of themselves, since $\Epsilon$ is an ordering relation on the ordinal classes. That is:

$\forall A \in \operatorname{On}: A \not\in A$ (1)

However, the class of all ordinal numbers is an ordinal itself. Since for ordering relations on the ordinals, the membership relation is equivalent to the subset relation in all instances (see the definition of Ordinals), we have that:

$\forall x \in \operatorname{On}: x \subset \operatorname{On}$ (2)

The segment of the class of ordinals is:

{ $x \in \operatorname{On} | x \subset \operatorname{On}$ } (3)

Which, by (2) is equal to the $\operatorname{On}$. Therefore $\operatorname{On}$ is an Ordinal.

Source

 * :$7.12$