If $A$ is a set and $A\subset A\times A$, then prove that $A=\varnothing$.
In the exercise, it says that I am supposed to use Zermelo-Fraenkel axioms to prove this statement.
I have tried to use Axiom of Regularity to an extent, but couldn't figure out anything from there to precede.
(Axiom of Regularity): $\forall x\,(x\neq \varnothing \rightarrow \exists y\in x\,(y\cap x=\varnothing ))$.
EDIT: I realize that a similar question has been asked but the answer I am looking for specifically involves only the Zermelo-Fraenkel axioms. So the other answer is not sufficient for me. Any contribution will be appreciated.