TITLE: Power set of $X$ is not in $X$
QUESTION [5 upvotes]: Prove or find a counterexample to: $P(X) \notin X$.

I think it's true but I'm having trouble coming up with a proof, clearly if $P(X) \in X$ then $X \in P(X) \in X$ but I'm not sure where the contradiction comes from. Does it require well-foundedness?

REPLY [4 votes]: Yes. This requires well-foundedness of the $\in$ relation. And that's exactly where the contradiction comes from. $\{X,\mathcal P(X)\}$ has no $\in$-minimal element.
It is consistent, however, that the axiom of foundation fails, and there is some $X$ such that $\mathcal P(X)\in X$. Note that it does not imply $\mathcal P(X)\subseteq X$, of course, since Cantor's theorem still holds.