TITLE: Two sets which contain each other as elements
QUESTION [6 upvotes]: In Zermelo-Fraenkel set theory do there exist sets $a,b$ such that $a \in b$ and $b \in a$. I think that no such sets exist but I am not sure how to prove why this is the case.

REPLY [4 votes]: This is not possible due to the axiom of regularity. Define $c = \{a,b\}$ and suppose $a\in b$ and $b\in a$. Then both $a$ and $c$ contain $b$ and thus the two are not disjoint. Similarly $a\in c\cap b$. Therefore $c$ does not contain a member that is disjoint with $c$, which means the axiom of regularity doesn't hold.