Ordinal Class is Ordinal

Theorem

The Ordinal Class $\On$ is an ordinal.

Proof

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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 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.

$\blacksquare$