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Theorem xnegmnf 13249
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 13152 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 11315 . . 3 -∞ ≠ +∞
3 ifnefalse 4543 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 5 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2735 . . 3 -∞ = -∞
65iftruei 4538 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2767 1 -𝑒-∞ = +∞
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wne 2938  ifcif 4531  +∞cpnf 11290  -∞cmnf 11291  -cneg 11491  -𝑒cxne 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-un 7754  ax-cnex 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-pnf 11295  df-mnf 11296  df-xr 11297  df-xneg 13152
This theorem is referenced by:  xnegcl  13252  xnegneg  13253  xltnegi  13255  xnegid  13277  xnegdi  13287  xsubge0  13300  xmulneg1  13308  xmulpnf1n  13317  xadddi2  13336  xrsdsreclblem  21448  xaddeq0  32764  xrge0npcan  33008  carsgclctunlem2  34301  supminfxr  45414  supminfxr2  45419  liminf0  45749  liminflbuz2  45771  liminfpnfuz  45772
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