If Set Exists then Empty Set Exists

Theorem
If at least one set exists, then there exists an empty set.

Proof
Let $S$ be a set.

By the axiom of class comprehension, there is an empty class:
 * $\O = \set { x : x \ne x }$

Since $x \in \O$ is never true, it follows vacuously that:
 * $x \in \O \implies x \in S$

By the subclass definition:
 * $\O \subseteq S$

By Subclass of Set is Set, $\O$ is a set.