Consider the claim:

(C) There is a transitive set $S \in V$ such that the structure $(S, \in)$ is an elementary submodel of $(V, \in)$.

Obviously, this claim cannot be a theoreom of ZFC, by Godel's 2nd Incompleteness Theorem. But does (C) follow from ZFC+CON(ZFC)? Or are large cardinals needed to prove (C)? More generally (and vaguely), what is the weakest assumption needed to prove (C)?