# Ordinal Class is Ordinal

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## 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:

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

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

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $7.12$