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Theorem elpqn 8791
Description: Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elpqn  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )

Proof of Theorem elpqn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq 8778 . . 3  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 ssrab2 3420 . . 3  |-  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. ) ( y  ~Q  x  ->  -.  ( 2nd `  x ) 
<N  ( 2nd `  y
) ) }  C_  ( N.  X.  N. )
31, 2eqsstri 3370 . 2  |-  Q.  C_  ( N.  X.  N. )
43sseli 3336 1  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725   A.wral 2697   {crab 2701   class class class wbr 4204    X. cxp 4867   ` cfv 5445   2ndc2nd 6339   N.cnpi 8708    <N clti 8711    ~Q ceq 8715   Q.cnq 8716
This theorem is referenced by:  nqereu  8795  nqerid  8799  enqeq  8800  addpqnq  8804  mulpqnq  8807  ordpinq  8809  addclnq  8811  mulclnq  8813  addnqf  8814  mulnqf  8815  adderpq  8822  mulerpq  8823  addassnq  8824  mulassnq  8825  distrnq  8827  mulidnq  8829  recmulnq  8830  ltsonq  8835  lterpq  8836  ltanq  8837  ltmnq  8838  ltexnq  8841  archnq  8846  wuncn  9034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-in 3319  df-ss 3326  df-nq 8778
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