Universal Class

Definition 1. The universal class is the class of all sets:

\[\V = \{x : x = x\}.\]

Proposition 1. $\V$ is a proper class.

Proof

Since $\V = \V$, it follows that $\V \in \V$, contrary to irreflexivity of $\in$.

References

[1]: Wikipedia contributors. Universal set. Wikipedia, The Free Encyclopedia.
[2]: Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 6.