Universal Class is Proper/Proof 1

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Theorem

Let $V$ denote the universal class.


Then $V$ is a proper class.


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.

$\blacksquare$


Sources