Universal Class is Proper

Theorem
Let $U$ denote the universal class.

$U$ is a proper class.

Proof
Assume that $U$ is small.

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

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

This contradicts Russell's Paradox.