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Can anyone explain what Mostowski collapse lemma is?

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).

What does this mean? This explanation seems very unclear to me.

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    Please add [the source of your quotation](http://en.wikipedia.org/wiki/Mostowski_collapse_lemma) when quoting.2012-05-31

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The Mostowski collapse lemma talks about well-founded relations. The nice thing about well-founded relations is that we can use them for sort of an induction.

The idea is that if $(A,R)$ is any well-founded and extensional partially ordered class, then there is a unique transitive $M$ such that $(A,R)\cong(M,\in)$.

We go about proving this by induction over $R$ we define $\pi(a)=\{\pi(b)\mid bRa\}$. So for minimal elements this is vacuously true (there can only be one, due to extensionality, and there is one, due to well-foundedness) and then we can easily step up by induction.