Theorem nqex 11047

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 11036	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 11005	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7799	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5309	1 ⊢ Q ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   class class class wbr 5108   × cxp 5656  ‘cfv 6566  2^nd c2nd 8034  Ncnpi 10968   <_N clti 10971   ~_Q ceq 10975  Qcnq 10976

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-om 7912  df-ni 10996  df-nq 11036

This theorem is used by:  npex  11110  elnp  11111  genpv  11123  genpdm  11126