Theorem nqex 11015

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 11004	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10973	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7771	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5302	1 ⊢ Q ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   class class class wbr 5102   × cxp 5649  ‘cfv 6543  2^nd c2nd 8004  Ncnpi 10936   <_N clti 10939   ~_Q ceq 10943  Qcnq 10944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7882  df-ni 10964  df-nq 11004

This theorem is used by:  npex  11078  elnp  11079  genpv  11091  genpdm  11094