TITLE: Is it possible that $x\in y$ and $y\in x$ in ZF?
QUESTION [5 upvotes]: In ZF, is it possible that $x\in y$ and $y\in x$? I don't know much about set theory, thanks for answers.

REPLY [6 votes]: No. If that were the case, the set $z=\lbrace x,y \rbrace$ would contradict the axiom of foundation. This argument also shows that in ZFC you can have no cycles.