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Theorem xnegmnf 12289
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 12192 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 10386 . . 3 -∞ ≠ +∞
3 ifnefalse 4290 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 5 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2800 . . 3 -∞ = -∞
65iftruei 4285 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2826 1 -𝑒-∞ = +∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wne 2972  ifcif 4278  +∞cpnf 10361  -∞cmnf 10362  -cneg 10558  -𝑒cxne 12189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-pow 5036  ax-un 7184  ax-cnex 10281
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-rex 3096  df-rab 3099  df-v 3388  df-un 3775  df-in 3777  df-ss 3784  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-uni 4630  df-pnf 10366  df-mnf 10367  df-xr 10368  df-xneg 12192
This theorem is referenced by:  xnegcl  12292  xnegneg  12293  xltnegi  12295  xnegid  12317  xnegdi  12326  xsubge0  12339  xmulneg1  12347  xmulpnf1n  12356  xadddi2  12375  xrsdsreclblem  20113  xaddeq0  30035  xrge0npcan  30209  carsgclctunlem2  30896  supminfxr  40432  supminfxr2  40437  liminf0  40764
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