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Theorem xnegpnf 9452
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 9400 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2100 . . 3 +∞ = +∞
32iftruei 3427 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2120 1 -𝑒+∞ = -∞
Colors of variables: wff set class
Syntax hints:   = wceq 1299  ifcif 3421  +∞cpnf 7669  -∞cmnf 7670  -cneg 7805  -𝑒cxne 9397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-if 3422  df-xneg 9400
This theorem is referenced by:  xnegcl  9456  xnegneg  9457  xltnegi  9459  xnegid  9483  xnegdi  9492  xaddass2  9494  xsubge0  9505  xposdif  9506  xlesubadd  9507  xblss2ps  12332  xblss2  12333
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