# Ordinal is Member of Ordinal Class

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## Theorem

Let $A$ be an ordinal.

Then:

- $A \in \operatorname{On} \lor A = \operatorname{On}$

where $\operatorname{On}$ denote the class of ordinals .

## Proof

By hypothesis $A$ is an ordinal

From Ordinal Class is Ordinal and Ordinal Membership is Trichotomy:

- $A \in \operatorname{On} \lor A = \operatorname{On} \lor \operatorname{On} \in A$

But by the Burali-Forti Paradox $\operatorname{On}$ is a proper class.

Therefore:

- $A \in \operatorname{On} \lor A = \operatorname{On}$

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $7.14$