Theorem nqex 10878

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 10867	. 2 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
2		niex 10836	. . 3 ⊢ N ∈ V
3	2, 2	xpex 7732	. 2 ⊢ (N × N) ∈ V
4	1, 3	rabex2 5296	1 ⊢ Q ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   class class class wbr 5099   × cxp 5643  ‘cfv 6517  2^nd c2nd 7965  Ncnpi 10799   <_N clti 10802   ~_Q ceq 10806  Qcnq 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-om 7843  df-ni 10827  df-nq 10867

This theorem is referenced by:  npex  10941  elnp  10942  genpv  10954  genpdm  10957