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Theorem 0nnq 8806
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 4909 . 2  |-  -.  (/)  e.  ( N.  X.  N. )
2 df-nq 8794 . . . 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 4215    X. cxp 4879   ` cfv 5457   2ndc2nd 6351   N.cnpi 8724    <N clti 8727    ~Q ceq 8731   Q.cnq 8732
This theorem is referenced by:  adderpq  8838  mulerpq  8839  addassnq  8840  mulassnq  8841  distrnq  8843  recmulnq  8846  recclnq  8848  ltanq  8853  ltmnq  8854  ltexnq  8857  nsmallnq  8859  ltbtwnnq  8860  ltrnq  8861  prlem934  8915  ltaddpr  8916  ltexprlem2  8919  ltexprlem3  8920  ltexprlem4  8921  ltexprlem6  8923  ltexprlem7  8924  prlem936  8929  reclem2pr  8930
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 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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 4270  df-xp 4887  df-nq 8794
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