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Theorem xnegpnf 9830
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 9774 . 2 -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2 eqid 2177 . . 3 +∞ = +∞
32iftruei 3542 . 2 if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
41, 3eqtri 2198 1 -𝑒+∞ = -∞
Colors of variables: wff set class
Syntax hints:   = wceq 1353  ifcif 3536  +∞cpnf 7991  -∞cmnf 7992  -cneg 8131  -𝑒cxne 9771
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 3537  df-xneg 9774
This theorem is referenced by:  xnegcl  9834  xnegneg  9835  xltnegi  9837  xnegid  9861  xnegdi  9870  xaddass2  9872  xsubge0  9883  xposdif  9884  xlesubadd  9885  xblss2ps  13943  xblss2  13944
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