Statement
Suppose that R is a binary relation on a class X such that
- R is set-like: R−1 = {y : y R x} is a set for every x,
- R is well-founded: every nonempty subset S of X contains an R-minimal element (i.e. an element x ∈ S such that R−1 ∩ S is empty),
- R is extensional: R−1 ≠ R−1 for every distinct elements x and y of X
The Mostowski collapse lemma states that for any 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. The isomorphism maps each element x of X to the set of images of elements y of X such that y R x (Jech 2003:69).
Read more about this topic: Mostowski Collapse Lemma
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