{{Short description|Transitive class including powersets of elements}} In set theory, a '''supertransitive class''' is a transitive class<ref>Any 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.}}</ref> 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 :<math>(\forall x)(x\in A \to \mathcal{P}(x) \subseteq A).</math><ref>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.}}</ref> Here ''P''(''x'') denotes the power set of ''x''.<ref>''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).</ref>
==See also== {{div col}} * Rank (set theory) {{div col end}}
==References== {{reflist}}
Category:Set theory