Working in $$\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V\_{j(\alpha)} \subsetneq V\_\alpha \land \alpha \text{ is limit }$$


Of course $j$ is *external* in the sense that it is not used in instances of separation nor replacement.


Now suppose that $S \subsetneq V\_\alpha$ and such that the ordinal rank of $S$ is $\alpha$, and such that $S$ is transitive; then by automorphism of $j$ we have $j(S) \subsetneq V\_{j(\alpha)}$, but:



> 
> is it always the case that $j(S) \subsetneq S$?
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> 
> if the answer is to the negative then what are the properties that a transitive proper subset of $V\_\alpha$ of rank $\alpha$ should fulfill in order for it to be moved by $j$ to a proper subset of itself?
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Would automorphisms cause nested subset-hood?
Regarding the first question, no:


Let $\beta=j(\alpha)$ and $\beta+\gamma=\alpha$. Define a version of the cumulative hierarchy with the empty set replaced with $\beta$. (That is, let $V\_0’=\beta$, $V\_1’=\{\beta\}$, and then $V\_{\xi+1}’=\mathcal{P}(V\_\xi’)\cup\{\beta\}\backslash\{\emptyset\}$ for $\xi>0$, and take unions at limits.) Let $S=V’\_{\gamma}$. This gives a counterexample.


Don’t know about the second question...


