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Theorem zfreg2 7976
Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7975) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)
Hypothesis
Ref Expression
zfreg2.1  |-  A  e. 
_V
Assertion
Ref Expression
zfreg2  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Distinct variable group:    x, A

Proof of Theorem zfreg2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 zfreg2.1 . . 3  |-  A  e. 
_V
21zfregcl 7974 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3747 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disjr 3810 . . 3  |-  ( ( A  i^i  x )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2905 . 2  |-  ( E. x  e.  A  ( A  i^i  x )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 266 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   _Vcvv 3058    i^i cin 3412   (/)c0 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-reg 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-in 3420  df-nul 3738
This theorem is referenced by:  zfregfr  7982
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