Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xnegpnf GIF version

Theorem xnegpnf 9642
 Description: Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
Assertion
Ref Expression
xnegpnf -𝑒+∞ = -∞

Proof of Theorem xnegpnf
StepHypRef Expression
1 df-xneg 9590 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2140 . . 3 +∞ = +∞
32iftruei 3485 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2161 1 -𝑒+∞ = -∞
 Colors of variables: wff set class Syntax hints:   = wceq 1332  ifcif 3479  +∞cpnf 7822  -∞cmnf 7823  -cneg 7959  -𝑒cxne 9587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480  df-xneg 9590 This theorem is referenced by:  xnegcl  9646  xnegneg  9647  xltnegi  9649  xnegid  9673  xnegdi  9682  xaddass2  9684  xsubge0  9695  xposdif  9696  xlesubadd  9697  xblss2ps  12613  xblss2  12614
 Copyright terms: Public domain W3C validator