Universal Class is Proper/Proof 1

Proof
Assume that $\mathrm U$ is small.

Note that $\operatorname{Ru} \subseteq \mathrm U$ where $\operatorname{Ru}$ denotes the Russell class.

By Axiom of Subsets Equivalents, $\operatorname{Ru}$ is also small.

This contradicts Russell's Paradox.