Theorem nqex 10844

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 10833	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10802	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7703	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5276	1 ⊢ Q ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   class class class wbr 5079   × cxp 5623  ‘cfv 6492  2^nd c2nd 7937  Ncnpi 10765   <_N clti 10768   ~_Q ceq 10772  Qcnq 10773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-om 7814  df-ni 10793  df-nq 10833

This theorem is referenced by:  npex  10907  elnp  10908  genpv  10920  genpdm  10923