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Theorem elpqn 8549
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 8536 . . 3  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 ssrab2 3258 . . 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 3208 . 2  |-  Q.  C_  ( N.  X.  N. )
43sseli 3176 1  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684   A.wral 2543   {crab 2547   class class class wbr 4023    X. cxp 4687   ` cfv 5255   2ndc2nd 6121   N.cnpi 8466    <N clti 8469    ~Q ceq 8473   Q.cnq 8474
This theorem is referenced by:  nqereu  8553  nqerid  8557  enqeq  8558  addpqnq  8562  mulpqnq  8565  ordpinq  8567  addclnq  8569  mulclnq  8571  addnqf  8572  mulnqf  8573  adderpq  8580  mulerpq  8581  addassnq  8582  mulassnq  8583  distrnq  8585  mulidnq  8587  recmulnq  8588  ltsonq  8593  lterpq  8594  ltanq  8595  ltmnq  8596  ltexnq  8599  archnq  8604  wuncn  8792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-in 3159  df-ss 3166  df-nq 8536
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