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Theorem nqex 9366
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 9355 . 2  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 niex 9324 . . 3  |-  N.  e.  _V
32, 2xpex 6614 . 2  |-  ( N. 
X.  N. )  e.  _V
41, 3rabex2 4552 1  |-  Q.  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1904   A.wral 2756   _Vcvv 3031   class class class wbr 4395    X. cxp 4837   ` cfv 5589   2ndc2nd 6811   N.cnpi 9287    <N clti 9290    ~Q ceq 9294   Q.cnq 9295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-om 6712  df-ni 9315  df-nq 9355
This theorem is referenced by:  npex  9429  elnp  9430  genpv  9442  genpdm  9445
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