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Theorem xnegmnf 13252
Description: Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegmnf -𝑒-∞ = +∞

Proof of Theorem xnegmnf
StepHypRef Expression
1 df-xneg 13154 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 11317 . . 3 -∞ ≠ +∞
3 ifnefalse 4537 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 5 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2737 . . 3 -∞ = -∞
65iftruei 4532 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2769 1 -𝑒-∞ = +∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wne 2940  ifcif 4525  +∞cpnf 11292  -∞cmnf 11293  -cneg 11493  -𝑒cxne 13151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pow 5365  ax-un 7755  ax-cnex 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-pnf 11297  df-mnf 11298  df-xr 11299  df-xneg 13154
This theorem is referenced by:  xnegcl  13255  xnegneg  13256  xltnegi  13258  xnegid  13280  xnegdi  13290  xsubge0  13303  xmulneg1  13311  xmulpnf1n  13320  xadddi2  13339  xrsdsreclblem  21430  xaddeq0  32757  xrge0npcan  33025  carsgclctunlem2  34321  supminfxr  45475  supminfxr2  45480  liminf0  45808  liminflbuz2  45830  liminfpnfuz  45831
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