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

Theorem xnegpnf 9828
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 9772 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2177 . . 3 +∞ = +∞
32iftruei 3541 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2198 1 -𝑒+∞ = -∞
Colors of variables: wff set class
Syntax hints:   = wceq 1353  ifcif 3535  +∞cpnf 7989  -∞cmnf 7990  -cneg 8129  -𝑒cxne 9769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3536  df-xneg 9772
This theorem is referenced by:  xnegcl  9832  xnegneg  9833  xltnegi  9835  xnegid  9859  xnegdi  9868  xaddass2  9870  xsubge0  9881  xposdif  9882  xlesubadd  9883  xblss2ps  13907  xblss2  13908
  Copyright terms: Public domain W3C validator