My understanding of Axiomatic Set theory is not too deep so apologies if its not very clear. This question is assuming we are considering ZFC.
Background to question
Consider the proposition $P$. For sets $x,y$ we say define recursively $ P(x,y)$ is true $ \iff (x \in y)\lor (\exists a \in y (P(x,a)))$
Alternatively $P(x,y)$ is true if $x\in y$ or $\exists$ z at the end of a terminating chain of inclusions of $y$ such that $x \in z$.
It seems intuitively true to me that $P(x,x)\to\bot$ but I'm not sure how to formalize it.
Assuming $P(x,x)$ to be true we see that evidentially $x\not \in x$ so we must have that $\exists$ some $z \in a \in b \in \cdots\in x$ such that $x \in z$. But this just means you can follow a chain of inclusions of x and find x in itself again.
It seems analogous to a case of sets $(a \in b) \land (b\in a)$ but two such sets can never exist as it contradicts the axiom of regularity.