Decide whether $\{ x | x=x\times x\}$ is a set

This is an exercise my set theory teacher suggested we try and I feel like the answer is yes (because it seems to me that no set other than $\emptyset$ satisfies $x\times x=x$, then the collection would be exactly $\{\emptyset\}$, which is a set), but I am stuck trying to prove that if $x\neq\emptyset$ then $x\neq x\times x$. I have tried using the axiom of regularity and cardinality supposing $x=x\times x$ but I reach no contradiction... could you give me any clues? (the theory is ZFC)