Ordinal Member of Ordinal Class
Theorem
Suppose $A$ is an ordinal. Then $A \in \operatorname{On} \lor A = \operatorname{On}$.
Proof
Since Ordinal Class is Ordinal, and $A$ is an ordinal, and Ordinal Membership Trichotomy, then $( A \in \operatorname{On} \lor A = \operatorname{On} \lor \operatorname{On} \in A )$. But $\operatorname{On}$ is a proper class by the Burali-Forti Paradox, so $( A \in \operatorname{On} \lor A = \operatorname{On}$ ). $\blacksquare$
please say or link to why being a proper class means that $\operatorname{On} \not\in A$
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