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Theorem elpqn 8833
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 8820 . . 3  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 ssrab2 3414 . . 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 3364 . 2  |-  Q.  C_  ( N.  X.  N. )
43sseli 3330 1  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1727   A.wral 2711   {crab 2715   class class class wbr 4237    X. cxp 4905   ` cfv 5483   2ndc2nd 6377   N.cnpi 8750    <N clti 8753    ~Q ceq 8757   Q.cnq 8758
This theorem is referenced by:  nqereu  8837  nqerid  8841  enqeq  8842  addpqnq  8846  mulpqnq  8849  ordpinq  8851  addclnq  8853  mulclnq  8855  addnqf  8856  mulnqf  8857  adderpq  8864  mulerpq  8865  addassnq  8866  mulassnq  8867  distrnq  8869  mulidnq  8871  recmulnq  8872  ltsonq  8877  lterpq  8878  ltanq  8879  ltmnq  8880  ltexnq  8883  archnq  8888  wuncn  9076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-in 3313  df-ss 3320  df-nq 8820
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