Supertransitive class

{{Short description|Transitive class including powersets of elements}}

In set theory, a supertransitive class is a transitive classAny element of a transitive set must also be its subset. See Definition 7.1 of {{cite book|last1=Zaring W.M.|first1= G. Takeuti|title=Introduction to axiomatic set theory|date=1971|publisher=Springer-Verlag|location=New York|isbn=0387900241|edition=2nd, rev.}} which includes as a subset the power set of each of its elements.

Formally, let A be a transitive class. Then A is supertransitive if and only if

:$(\backslash forall\; x)(x\backslash in\; A\; \backslash to\; \backslash mathcal\{P\}(x)\; \backslash subseteq\; A).$See Definition 9.8 of {{cite book|last1=Zaring W.M.|first1= G. Takeuti|title=Introduction to axiomatic set theory|date=1971|publisher=Springer-Verlag|location=New York|isbn=0387900241|edition=2nd, rev.}}

Here P(x) denotes the power set of x.P(x) must be a set by axiom of power set, since each element x of a class A must be a set (Theorem 4.6 in Takeuti's text above).