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Theorem xnegmnf 12079
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 11984 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 10133 . . 3 -∞ ≠ +∞
3 ifnefalse 4131 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 5 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2651 . . 3 -∞ = -∞
65iftruei 4126 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2677 1 -𝑒-∞ = +∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wne 2823  ifcif 4119  +∞cpnf 10109  -∞cmnf 10110  -cneg 10305  -𝑒cxne 11981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-rex 2947  df-rab 2950  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-pnf 10114  df-mnf 10115  df-xr 10116  df-xneg 11984
This theorem is referenced by:  xnegcl  12082  xnegneg  12083  xltnegi  12085  xnegid  12107  xnegdi  12116  xsubge0  12129  xmulneg1  12137  xmulpnf1n  12146  xadddi2  12165  xrsdsreclblem  19840  xaddeq0  29646  xrge0npcan  29822  carsgclctunlem2  30509  supminfxr  40007  supminfxr2  40012  liminf0  40343
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