Theorem nqex 10964

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 10953	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10922	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7774	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5340	1 ⊢ Q ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   class class class wbr 5142   × cxp 5682  ‘cfv 6560  2^nd c2nd 8014  Ncnpi 10885   <_N clti 10888   ~_Q ceq 10892  Qcnq 10893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-om 7889  df-ni 10913  df-nq 10953

This theorem is referenced by:  npex  11027  elnp  11028  genpv  11040  genpdm  11043