Set Consisting of Empty Set is not Empty

Theorem
Let $S$ be the set defined as:
 * $S = \set \O$

Then $S$ is not the empty set.

That is:
 * $\O \ne \set \O$

Proof
We have:
 * $\O \in \set \O$

and so:
 * $\neg \paren {\forall x: x \notin \O}$

The result follows by definition of the empty set.