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Theorem nqex 8563
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 8552 . 2  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 niex 8521 . . . 4  |-  N.  e.  _V
32, 2xpex 4817 . . 3  |-  ( N. 
X.  N. )  e.  _V
43rabex 4181 . 2  |-  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. ) ( y  ~Q  x  ->  -.  ( 2nd `  x ) 
<N  ( 2nd `  y
) ) }  e.  _V
51, 4eqeltri 2366 1  |-  Q.  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039    X. cxp 4703   ` cfv 5271   2ndc2nd 6137   N.cnpi 8482    <N clti 8485    ~Q ceq 8489   Q.cnq 8490
This theorem is referenced by:  npex  8626  elnp  8627  genpv  8639  genpdm  8642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-ni 8512  df-nq 8552
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