Universal Class is Proper

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Let $V$ denote the universal class.

Then $V$ is a proper class.

Proof 1

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.


Proof 2


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Let $I_\mathrm U$ be the identity mapping on $\mathrm U$.

By Identity Mapping is Bijection it follows that $I_\mathrm U$ is a bijection.

Therefore, by the Axiom of Limitation of Size, $\mathrm U$ is proper.