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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.
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I think we'd also have to suppose P(Ø) to conclude P(A) holds for all sets.

Otherwise we could prove any P that's always false holds for all sets.

--Dan