This is the so-called epsilon induction, and it requires not the Axiom of Choice but the Axiom of Restriction. Suppose that for any set A, if P(a) holds for each a in A, then P(A) holds. Then P(A) holds for all sets. The theories of transfinite induction and well-founded relations both generalize induction; see Thomas Jech's book on set theory. Bob Mayans