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Theorem 0nnq 8564
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 4733 . 2  |-  -.  (/)  e.  ( N.  X.  N. )
2 df-nq 8552 . . . 4  |-  Q.  =  { y  e.  ( N.  X.  N. )  |  A. x  e.  ( N.  X.  N. )
( y  ~Q  x  ->  -.  ( 2nd `  x
)  <N  ( 2nd `  y
) ) }
3 ssrab2 3271 . . . 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 3221 . . 3  |-  Q.  C_  ( N.  X.  N. )
54sseli 3189 . 2  |-  ( (/)  e.  Q.  ->  (/)  e.  ( N.  X.  N. )
)
61, 5mto 167 1  |-  -.  (/)  e.  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696   A.wral 2556   {crab 2560   (/)c0 3468   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:  adderpq  8596  mulerpq  8597  addassnq  8598  mulassnq  8599  distrnq  8601  recmulnq  8604  recclnq  8606  ltanq  8611  ltmnq  8612  ltexnq  8615  nsmallnq  8617  ltbtwnnq  8618  ltrnq  8619  prlem934  8673  ltaddpr  8674  ltexprlem2  8677  ltexprlem3  8678  ltexprlem4  8679  ltexprlem6  8681  ltexprlem7  8682  prlem936  8687  reclem2pr  8688
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-nq 8552
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