Suppose that R is a binary relation on a class X such that The Mostowski collapse lemma states that for every such R there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (X, R), and the isomorphism is unique.
This implies the following variant of the Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.
The well-foundedness assumption of the Mostowski lemma can be alleviated or dropped in non-well-founded set theories.
In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class.
In set theory with Aczel's anti-foundation axiom, every set-like relation is bisimilar to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.