# 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.

$\blacksquare$