What is the consistency strength of the following prinicple?


Every set is a member of a [supertransitive](https://en.wikipedia.org/wiki/Supertransitive_class) model of $\sf ZFC$.


Formally this means adding the following sentence to axioms of $\sf ZFC$:


$\forall x \exists M :( M \models {\sf ZFC}) \land \operatorname {supertransitive}(M) \land x \in M$


Is it equivalent to having a proper class of worldly cardinals?



What is the strength of having all sets being elements of supertransitive models of ZFC?
Every level of the cumulative hierarchy is supertransitive, so "There is a proper class of worldly cardinals" clearly implies the principle in question.


Conversely, if $M\models\mathsf{ZFC}$ is supertransitive then $M$ computes powersets correctly: if $x\in M$ then $\mathcal{P}(x)\subseteq M$ so $\mathcal{P}(x)^M=\mathcal{P}(x)$. Consequently, every supertransitive model of $\mathsf{ZFC}$ is some $V\_\alpha$.


So the answer to your question is **yes**, by virtue of the fact that the supertransitive models of $\mathsf{ZFC}$ are essentially-by-definition the worldly levels of the cumulative hierarchy.


