Set is Not Element of Itself

Theorem
There cannot exist a set which is an element of itself.

That is:
 * $\neg \exists a: a \in a$

Proof
$a$ is such a set.

Then $a \in a$ and $a \in \set a$.

$a \ne \O$ because the empty set has no elements by definition.

It is also seen that:

By the Axiom of Foundation:
 * $a \cap \set a = \O$

Thus $a = \O$.

But it was previously established that $a \ne \O$.

From this contradiction it follows that there cannot exist such a set.

Hence the result.