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Theorem zfreg2 8120
Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 8119) 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 8118 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3771 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disjr 3836 . . 3  |-  ( ( A  i^i  x )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2924 . 2  |-  ( E. x  e.  A  ( A  i^i  x )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 269 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    i^i cin 3435   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-reg 8116
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-in 3443  df-nul 3762
This theorem is referenced by:  zfregfr  8126
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