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
Aiming for contradiction, let $a$ be such a set.

Then $a \in a$ and $a \in \left\{{a}\right\}$.

$a \ne \varnothing$ because the empty set doesn't have any elements.

It is also seen that:

By the Axiom of Foundation, it follows that $a \cap \left\{{a}\right\} = \varnothing$.

And so $a = \varnothing$.

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

This is a contradiction, therefore there cannot exist such a set.

Hence the result.