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Theorem nqex 8547
Description: The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqex  |-  Q.  e.  _V

Proof of Theorem nqex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq 8536 . 2  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 niex 8505 . . . 4  |-  N.  e.  _V
32, 2xpex 4801 . . 3  |-  ( N. 
X.  N. )  e.  _V
43rabex 4165 . 2  |-  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. ) ( y  ~Q  x  ->  -.  ( 2nd `  x ) 
<N  ( 2nd `  y
) ) }  e.  _V
51, 4eqeltri 2353 1  |-  Q.  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023    X. cxp 4687   ` cfv 5255   2ndc2nd 6121   N.cnpi 8466    <N clti 8469    ~Q ceq 8473   Q.cnq 8474
This theorem is referenced by:  npex  8610  elnp  8611  genpv  8623  genpdm  8626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-ni 8496  df-nq 8536
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