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Axiom ax-reg 8039
Description: Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 8042) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8044). A stronger version that works for proper classes is proved as zfregs 8184. (Contributed by NM, 14-Aug-1993.)
Assertion
Ref Expression
ax-reg
Distinct variable group:   , ,

Detailed syntax breakdown of Axiom ax-reg
StepHypRef Expression
1 vy . . . 4
2 vx . . . 4
31, 2wel 1819 . . 3
43, 1wex 1612 . 2
5 vz . . . . . . 7
65, 1wel 1819 . . . . . 6
75, 2wel 1819 . . . . . . 7
87wn 3 . . . . . 6
96, 8wi 4 . . . . 5
109, 5wal 1393 . . . 4
113, 10wa 369 . . 3
1211, 1wex 1612 . 2
134, 12wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  axreg2  8040
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