# Universal Class is Proper

## Theorem

Let $\mathrm U$ denote the universal class.

Then $\mathrm U$ 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.

$\blacksquare$

## 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.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.23$