MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nnq Unicode version

Theorem 0nnq 9323
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

Proof of Theorem 0nnq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nelxp 5032 . 2
2 df-nq 9311 . . . 4
3 ssrab2 3584 . . . 4
42, 3eqsstri 3533 . . 3
54sseli 3499 . 2
61, 5mto 176 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  e.wcel 1818  A.wral 2807  {crab 2811   c0 3784   class class class wbr 4452  X.cxp 5002  `cfv 5593   c2nd 6799   cnpi 9243   clti 9246   ceq 9250   cnq 9251
This theorem is referenced by:  adderpq  9355  mulerpq  9356  addassnq  9357  mulassnq  9358  distrnq  9360  recmulnq  9363  recclnq  9365  ltanq  9370  ltmnq  9371  ltexnq  9374  nsmallnq  9376  ltbtwnnq  9377  ltrnq  9378  prlem934  9432  ltaddpr  9433  ltexprlem2  9436  ltexprlem3  9437  ltexprlem4  9438  ltexprlem6  9440  ltexprlem7  9441  prlem936  9446  reclem2pr  9447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010  df-nq 9311
  Copyright terms: Public domain W3C validator