Existence of Class Complement

Theorem

Let $X$ be a class.

Then there is a unique class $Z$ such that, for every set $u$:

$u \in Z \iff u \notin X$

Proof

Let $X$ be arbitrary.

By Axiom $\text B 3$, there is a class $Z$ such that:

$\forall u: u \in Z \iff u \notin X$

This satisfies the existence portion of the theorem.

$\Box$

Let $Z'$ be a class such that:

$\forall u: u \in Z' \iff u \notin X$

Then, for every set $u$:

$u \in Z \iff u \notin X \iff u \in Z'$

Therefore, by the Axiom of Extension:

$Z = Z'$

satisfying the uniqueness portion of the theorem.

$\blacksquare$