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From: zuhair on 18 Jun 2010 22:36
After I made a reformulation of Z-Reg. as I presented in the post to
this Usenet under the title_ A=Z-Regularity. The following is a
parallel formulation but of ZF-Regularity instead of Z-Reg.

A" is the set of all sentences (entailed from FOL with identity and
membership) by the following non logical axioms:

(1) unique replacement:For any formula phi(x,y) not containing the
letter B, the following is an axiom:

Ar[(Ax c r)AyAz(phi(x,y) & phi(x,z) -> y = z) ->
E!BAy(y e B <-> (Ex c r)phi(x,y))].

were c is the subset relation defined in the standard manner.

(2) Union: as in Z.
(3) Infinity: as in Z.

/Theory definition finished.

Although I don't have all the formal proofs of this, but I do think

A=ZF-Reg.

If I am correct then this would be another short reformulation of ZF-
Reg. to only three axiom schemes instead of the standard five axiom
schemes.

Zuhair

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