Geodesic
Foundations of a new axiomatic set theory
Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 467-473

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A new axiomatic set theory, consisting of four axioms, is presented. In this theory one can prove as theorems all of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), except for the axiom of regularity. 
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     author = {A. M. Vdovin},
     title = {Foundations of a new axiomatic set theory},
     journal = {Izvestiya. Mathematics },
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A. M. Vdovin. Foundations of a new axiomatic set theory. Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 467-473. http://geodesic.mathdoc.fr/item/IM2_1991_37_2_a11/