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Theorem nqex 10961
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 ∈ V

Proof of Theorem nqex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq 10950 . 2 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
2 niex 10919 . . 3 N ∈ V
32, 2xpex 7772 . 2 (N × N) ∈ V
41, 3rabex2 5347 1 Q ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  wral 3059  Vcvv 3478   class class class wbr 5148   × cxp 5687  cfv 6563  2nd c2nd 8012  Ncnpi 10882   <N clti 10885   ~Q ceq 10889  Qcnq 10890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888  df-ni 10910  df-nq 10950
This theorem is referenced by:  npex  11024  elnp  11025  genpv  11037  genpdm  11040
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