Universal Class is Proper/Proof 2

Theorem

Let $V$ denote the universal class.

Then $V$ is a proper class.

Proof

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Let $I_V$ be the identity mapping on $V$.

By Identity Mapping is Bijection it follows that $I_V$ is a bijection.

Therefore, by the Axiom of Limitation of Size, $V$ is proper.

$\blacksquare$