Theorem nqex 10811

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 10800	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10769	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7686	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5279	1 ⊢ Q ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   class class class wbr 5091   × cxp 5614  ‘cfv 6481  2^nd c2nd 7920  Ncnpi 10732   <_N clti 10735   ~_Q ceq 10739  Qcnq 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-om 7797  df-ni 10760  df-nq 10800

This theorem is referenced by:  npex  10874  elnp  10875  genpv  10887  genpdm  10890