I will follow James M Henle'sAn Outline of Set Theory.

The Axiom of Regularity: If $D$ is a set, then either $D=\emptyset$ or else & $(\exists B\in D)[D\cap B=\emptyset]$

Now suppose that $(\exists A)[A\in A]$ Define $D=\{A\}$. Clearly that is a non-empty set.

So by the axiom $(\exists B\in D)[D\cap B=\emptyset]$ but $D$ contains only one element so $B=A$.

But that means $A\in (A\cap B) \text{ or }A\in (D\cap B)\ne\emptyset$.