Supertransitive class

In set theory, a supertransitive class is a transitive class ^[1] 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

(\forall x)(x\in A\to {\mathcal {P}}(x)\subseteq A).

^[2]

Here P(x) denotes the power set of x.^[3]

References

^ Any element of a transitive set must also be its subset. See Definition 7.1 of Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York: Springer-Verlag. ISBN 0387900241.
^ See Definition 9.8 of Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York: Springer-Verlag. ISBN 0387900241.
^ 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).