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Theorem xnegmnf 8813
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 8760 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 8769 . . 3 -∞ ≠ +∞
3 ifnefalse 3367 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 7 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2054 . . 3 -∞ = -∞
65iftruei 3362 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2078 1 -𝑒-∞ = +∞
Colors of variables: wff set class
Syntax hints:   = wceq 1257  wne 2218  ifcif 3356  +∞cpnf 7086  -∞cmnf 7087  -cneg 7216  -𝑒cxne 8757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-un 4195  ax-cnex 7003
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-nel 2313  df-rex 2327  df-rab 2330  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-if 3357  df-pw 3386  df-sn 3406  df-pr 3407  df-uni 3606  df-pnf 7091  df-mnf 7092  df-xr 7093  df-xneg 8760
This theorem is referenced by:  xnegcl  8816  xnegneg  8817  xltnegi  8819
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