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Theorem xnegmnf 13189
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 13092 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 11270 . . 3 -∞ ≠ +∞
3 ifnefalse 4541 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 5 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2733 . . 3 -∞ = -∞
65iftruei 4536 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2765 1 -𝑒-∞ = +∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wne 2941  ifcif 4529  +∞cpnf 11245  -∞cmnf 11246  -cneg 11445  -𝑒cxne 13089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-pow 5364  ax-un 7725  ax-cnex 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-pnf 11250  df-mnf 11251  df-xr 11252  df-xneg 13092
This theorem is referenced by:  xnegcl  13192  xnegneg  13193  xltnegi  13195  xnegid  13217  xnegdi  13227  xsubge0  13240  xmulneg1  13248  xmulpnf1n  13257  xadddi2  13276  xrsdsreclblem  20991  xaddeq0  31966  xrge0npcan  32195  carsgclctunlem2  33318  supminfxr  44174  supminfxr2  44179  liminf0  44509  liminflbuz2  44531  liminfpnfuz  44532
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