Ordinal is Member of Ordinal Class

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Theorem

Let $A$ be an ordinal.


Then:

$A \in \On \lor A = \On$

where $\On$ denote the class of ordinals .


Proof

By hypothesis $A$ is an ordinal

From Ordinal Class is Ordinal and Ordinal Membership is Trichotomy:

$A \in \On \lor A = \On \lor \On \in A$

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

Therefore:

$A \in \On \lor A = \On$

$\blacksquare$



Sources