# Universal Class is Proper/Proof 2

## Theorem

Let $V$ denote the universal class.

Then $V$ is a proper class.

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

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.
$\blacksquare$