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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](http://en.wikipedia.org/wiki/Axiom_of_regularity). See also [this thread](http://math.stackexchange.com/q/125385/5363) for links and elaborations.2012-04-02

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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.