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Theorem nqex 9249
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 9238 . 2  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 niex 9207 . . 3  |-  N.  e.  _V
32, 2xpex 6540 . 2  |-  ( N. 
X.  N. )  e.  _V
41, 3rabex2 4544 1  |-  Q.  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1840   A.wral 2751   _Vcvv 3056   class class class wbr 4392    X. cxp 4938   ` cfv 5523   2ndc2nd 6735   N.cnpi 9170    <N clti 9173    ~Q ceq 9177   Q.cnq 9178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-om 6637  df-ni 9198  df-nq 9238
This theorem is referenced by:  npex  9312  elnp  9313  genpv  9325  genpdm  9328
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