Ordinal is Member of Class of All Ordinals

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:
 * $A \in \On \lor A = \On$