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Theorem 0nnq 8757
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 4865 . 2  |-  -.  (/)  e.  ( N.  X.  N. )
2 df-nq 8745 . . . 4  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
3 ssrab2 3388 . . . 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 3338 . . 3  |-  Q.  C_  ( N.  X.  N. )
54sseli 3304 . 2  |-  ( (/)  e.  Q.  ->  (/)  e.  ( N.  X.  N. )
)
61, 5mto 169 1  |-  -.  (/)  e.  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1721   A.wral 2666   {crab 2670   (/)c0 3588   class class class wbr 4172    X. cxp 4835   ` cfv 5413   2ndc2nd 6307   N.cnpi 8675    <N clti 8678    ~Q ceq 8682   Q.cnq 8683
This theorem is referenced by:  adderpq  8789  mulerpq  8790  addassnq  8791  mulassnq  8792  distrnq  8794  recmulnq  8797  recclnq  8799  ltanq  8804  ltmnq  8805  ltexnq  8808  nsmallnq  8810  ltbtwnnq  8811  ltrnq  8812  prlem934  8866  ltaddpr  8867  ltexprlem2  8870  ltexprlem3  8871  ltexprlem4  8872  ltexprlem6  8874  ltexprlem7  8875  prlem936  8880  reclem2pr  8881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-nq 8745
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