Theorem nqex 10882

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.

Step	Hyp	Ref	Expression
1		df-nq 10871	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10840	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7731	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5298	1 ⊢ Q ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   class class class wbr 5109   × cxp 5638  ‘cfv 6513  2^nd c2nd 7969  Ncnpi 10803   <_N clti 10806   ~_Q ceq 10810  Qcnq 10811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-inf2 9600

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-tr 5217  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-om 7845  df-ni 10831  df-nq 10871

This theorem is referenced by:  npex  10945  elnp  10946  genpv  10958  genpdm  10961