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Theorem 0nnq 9108
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 4882 . 2  |-  -.  (/)  e.  ( N.  X.  N. )
2 df-nq 9096 . . . 4  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
3 ssrab2 3452 . . . 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 3401 . . 3  |-  Q.  C_  ( N.  X.  N. )
54sseli 3367 . 2  |-  ( (/)  e.  Q.  ->  (/)  e.  ( N.  X.  N. )
)
61, 5mto 176 1  |-  -.  (/)  e.  Q.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1756   A.wral 2730   {crab 2734   (/)c0 3652   class class class wbr 4307    X. cxp 4853   ` cfv 5433   2ndc2nd 6591   N.cnpi 9026    <N clti 9029    ~Q ceq 9033   Q.cnq 9034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-opab 4366  df-xp 4861  df-nq 9096
This theorem is referenced by:  adderpq  9140  mulerpq  9141  addassnq  9142  mulassnq  9143  distrnq  9145  recmulnq  9148  recclnq  9150  ltanq  9155  ltmnq  9156  ltexnq  9159  nsmallnq  9161  ltbtwnnq  9162  ltrnq  9163  prlem934  9217  ltaddpr  9218  ltexprlem2  9221  ltexprlem3  9222  ltexprlem4  9223  ltexprlem6  9225  ltexprlem7  9226  prlem936  9231  reclem2pr  9232
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