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Theorem zfreg 7599
Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that  A be a set, that can be proved with more difficulty (see zfregs 7704). (Contributed by NM, 26-Nov-1995.)
Hypothesis
Ref Expression
zfreg.1  |-  A  e. 
_V
Assertion
Ref Expression
zfreg  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Distinct variable group:    x, A

Proof of Theorem zfreg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 zfreg.1 . . 3  |-  A  e. 
_V
21zfregcl 7598 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3625 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disj 3696 . . 3  |-  ( ( x  i^i  A )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2737 . 2  |-  ( E. x  e.  A  ( x  i^i  A )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 259 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1551    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   E.wrex 2713   _Vcvv 2965    i^i cin 3308   (/)c0 3616
This theorem is referenced by:  inf3lem3  7621  en3lp  7708  setindtr  27207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-reg 7596
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-v 2967  df-dif 3312  df-in 3316  df-nul 3617
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