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Theorem elpqn 8565
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 8552 . . 3  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
2 ssrab2 3271 . . 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 3221 . 2  |-  Q.  C_  ( N.  X.  N. )
43sseli 3189 1  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696   A.wral 2556   {crab 2560   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:  nqereu  8569  nqerid  8573  enqeq  8574  addpqnq  8578  mulpqnq  8581  ordpinq  8583  addclnq  8585  mulclnq  8587  addnqf  8588  mulnqf  8589  adderpq  8596  mulerpq  8597  addassnq  8598  mulassnq  8599  distrnq  8601  mulidnq  8603  recmulnq  8604  ltsonq  8609  lterpq  8610  ltanq  8611  ltmnq  8612  ltexnq  8615  archnq  8620  wuncn  8808
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-in 3172  df-ss 3179  df-nq 8552
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