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Theorem zfreg 8113
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 8218). (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 8112 . 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 disj 3833 . . 3  |-  ( ( x  i^i  A )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2927 . 2  |-  ( E. x  e.  A  ( x  i^i  A )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 269 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437   E.wex 1659    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    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 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-reg 8110
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-v 3083  df-dif 3439  df-in 3443  df-nul 3762
This theorem is referenced by:  en3lp  8124  inf3lem3  8138  setindtr  35799
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