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Is it possible in ZFC that you have a cyclic containment of sets, e.g., a inclusion like $A \in B$ and $B \in A$?

I never took set theory classes, I am just curious.

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No, this is excluded by the axiom of foundation. See also this thread for links and elaborations. – t.b. Apr 2 '12 at 17:12
up vote 3 down vote accepted

As mentioned in the comments, the impossibility of this (in ZF) follows from Axiom of regularity.

Every non-empty set $X$ contains an element $Y$ which is disjoint from $X$.

Suppose that $A\in B$ and $B\in A$. Define $X=\{A,B\}$. Then $A\in X\cap B$ and $B\in X\cap A$, i.e. for each element $Y$ of $X$ the intersection $Y\cap X$ is non-empty. This contradicts the Axiom of regularity.

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