Universal Class is Proper/Proof 3

Theorem

Let $V$ denote the universal class.

Then $V$ is a proper class.

Proof

Aiming for a contradiction, suppose $V$ is small.

By Cantor's Theorem, there is no surjection from $V$ to $\powerset V$.

By definition of universal class:

$\powerset V \subseteq V$

By Injection from Subset to Superset, there exists an injection from $\powerset V$ to $V$.

By Injection has Surjective Left Inverse Mapping, there is a surjection from $V$ to $\powerset V$.

But this contradicts Cantor's Theorem.

Therefore $V$ is proper.

$\blacksquare$