Class of All Ordinals is Proper Class
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Let $\On$ denote the class of all ordinals.
Then $\On$ is a proper class.
That is, $\On$ is not a set.
We have that Successor Mapping on Ordinals is Strictly Progressing.
The result follows from Superinductive Class under Strictly Progressing Mapping is Proper Class.
Aiming for a contradiction, suppose $\On$ is a set.
Then from the Burali-Forti Paradox, a contradiction could be deduced.
Hence by Proof by Contradiction, $\On$ cannot be a set.
Thus $\On$ is a class that is not a set.
Hence $\On$ is a proper class.