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Theorem xnegmnf 9505
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 9452 . 2 -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞))
2 mnfnepnf 7745 . . 3 -∞ ≠ +∞
3 ifnefalse 3451 . . 3 (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞))
42, 3ax-mp 7 . 2 if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)
5 eqid 2115 . . 3 -∞ = -∞
65iftruei 3446 . 2 if(-∞ = -∞, +∞, --∞) = +∞
71, 4, 63eqtri 2139 1 -𝑒-∞ = +∞
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wne 2282  ifcif 3440  +∞cpnf 7721  -∞cmnf 7722  -cneg 7857  -𝑒cxne 9449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-un 4315  ax-cnex 7636
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-rex 2396  df-rab 2399  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-uni 3703  df-pnf 7726  df-mnf 7727  df-xr 7728  df-xneg 9452
This theorem is referenced by:  xnegcl  9508  xnegneg  9509  xltnegi  9511  xnegid  9535  xnegdi  9544  xsubge0  9557  xposdif  9558
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