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Theorem zfreg2 7520
Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7519) 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 7518 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3597 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disjr 3629 . . 3  |-  ( ( A  i^i  x )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2691 . 2  |-  ( E. x  e.  A  ( A  i^i  x )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 258 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279   (/)c0 3588
This theorem is referenced by:  zfregfr  7526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-reg 7516
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589
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