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Theorem 0nnq 8803
Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
0nnq  |-  -.  (/)  e.  Q.

Proof of Theorem 0nnq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nelxp 4908 . 2  |-  -.  (/)  e.  ( N.  X.  N. )
2 df-nq 8791 . . . 4  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
3 ssrab2 3430 . . . 4  |-  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. ) ( y  ~Q  x  ->  -.  ( 2nd `  x ) 
<N  ( 2nd `  y
) ) }  C_  ( N.  X.  N. )
42, 3eqsstri 3380 . . 3  |-  Q.  C_  ( N.  X.  N. )
54sseli 3346 . 2  |-  ( (/)  e.  Q.  ->  (/)  e.  ( N.  X.  N. )
)
61, 5mto 170 1  |-  -.  (/)  e.  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1726   A.wral 2707   {crab 2711   (/)c0 3630   class class class wbr 4214    X. cxp 4878   ` cfv 5456   2ndc2nd 6350   N.cnpi 8721    <N clti 8724    ~Q ceq 8728   Q.cnq 8729
This theorem is referenced by:  adderpq  8835  mulerpq  8836  addassnq  8837  mulassnq  8838  distrnq  8840  recmulnq  8843  recclnq  8845  ltanq  8850  ltmnq  8851  ltexnq  8854  nsmallnq  8856  ltbtwnnq  8857  ltrnq  8858  prlem934  8912  ltaddpr  8913  ltexprlem2  8916  ltexprlem3  8917  ltexprlem4  8918  ltexprlem6  8920  ltexprlem7  8921  prlem936  8926  reclem2pr  8927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-nq 8791
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