The collapsing lemma, sometimes called *Mostowski-Shepherdson collapsing lemma* or the *Mostowsky collapse* is the following: **Theorem (The Collapsing Lemma).** Let $X$ be an *extensional set*, i.e. $ (\forall u, v \in X)(u \neq v \rightarrow(\exists x \in X)(x \in u \leftrightarrow x \notin v)) $Then there is a unique transitive set $M$ and a unique bijection $\pi: X \leftrightarrow M$ such that $ \pi:\langle X, \epsilon\rangle \cong\langle M, \epsilon\rangle $ Moreover, if $Y \subseteq X$ is transitive, then $\pi \upharpoonright Y=\mathrm{id} \upharpoonright Y$. The transitive set $M$ is called the *transitive collapse* or *transitivisation* of $X$.