Existence of Universal Class

Theorem

There exists a unique class $V$ such that:

$\forall u: u \in V$

That is, the universal class exists and is unique.

Proof

By Existence of Empty Class, there is a unique class $\O$ such that:

$\forall u: u \notin \O$

for every set $u$.

Thus, by Existence of Class Complement, there is a unique class $\overline \O$ such that:

$u \in \overline \O \iff u \notin \O$

But then, by the definition of $\O$:

$\forall u: u \in \overline \O$

Then, $V := \overline \O$ satisfies the theorem, and is unique.

$\blacksquare$