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Theorem nqex 9207
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.  e.  _V

Proof of Theorem nqex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq 9196 . 2  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 niex 9165 . . . 4  |-  N.  e.  _V
32, 2xpex 6621 . . 3  |-  ( N. 
X.  N. )  e.  _V
43rabex 4554 . 2  |-  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. ) ( y  ~Q  x  ->  -.  ( 2nd `  x ) 
<N  ( 2nd `  y
) ) }  e.  _V
51, 4eqeltri 2538 1  |-  Q.  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078   class class class wbr 4403    X. cxp 4949   ` cfv 5529   2ndc2nd 6689   N.cnpi 9126    <N clti 9129    ~Q ceq 9133   Q.cnq 9134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-om 6590  df-ni 9156  df-nq 9196
This theorem is referenced by:  npex  9270  elnp  9271  genpv  9283  genpdm  9286
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