Proof of every cofinal subclass of $\mathbf{ON}$ is proper Can you please tell me if my proof of the following claim is correct? Thank you! Claim: Every cofinal subclass of $\mathbf{ON}$ is proper. Proof: Let $A \subset \mathbf{ON}$ be cofinal. Assume $A$ was a set. Then $\bigcup A$ would be an ordinal exceeding all $a$ in $A$ hence contradicting cofinality of $A$. Hence $A$ must be a proper class.

It's fine. Another argument would be that every set of ordinals has a rank $\alpha$ so it is a subset of $\alpha$, and therefore not cofinal in $\mathbf{ON}$.