Editing Mostowski collapse lemma (section)

==Application==
Every set [[model theory|model]] of [[Zermelo–Fraenkel set theory|ZF]] is set-like and extensional. If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a [[transitive model]] of ZF and such a transitive model is unique.

Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the [[axiom of regularity]] is true in the model. There exists a model ''M'' (assuming the consistency of ZF) whose domain has a subset ''A'' with no ''R''-minimal element, but this set ''A'' is not a "set in the model" (''A'' is not in the domain of the model, even though all of its members are). More precisely, for no such set ''A'' there exists ''x'' in ''M'' such that ''A'' = ''R''<sup>−1</sup>[''x'']. So ''M'' satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it.