Universal Class is Proper/Proof 1

Theorem

Let $V$ denote the universal class.

Then $V$ is a proper class.

Proof

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

We have that:

$\operatorname {Ru} \subseteq V$

where $\operatorname {Ru}$ denotes the Russell class.

By Axiom of Subsets Equivalents, $\operatorname {Ru}$ is also small.

This contradicts Russell's Paradox.

$\blacksquare$

Sources

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