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Theorem xnegmnf 9861
Description: Minus -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegmnf  |-  -e -oo  = +oo

Proof of Theorem xnegmnf
StepHypRef Expression
1 df-xneg 9804 . 2  |-  -e -oo  =  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )
2 mnfnepnf 8044 . . 3  |- -oo  =/= +oo
3 ifnefalse 3560 . . 3  |-  ( -oo  =/= +oo  ->  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )  =  if ( -oo  = -oo , +oo ,  -u -oo )
)
42, 3ax-mp 5 . 2  |-  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )  =  if ( -oo  = -oo , +oo ,  -u -oo )
5 eqid 2189 . . 3  |- -oo  = -oo
65iftruei 3555 . 2  |-  if ( -oo  = -oo , +oo ,  -u -oo )  = +oo
71, 4, 63eqtri 2214 1  |-  -e -oo  = +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1364    =/= wne 2360   ifcif 3549   +oocpnf 8020   -oocmnf 8021   -ucneg 8160    -ecxne 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-un 4451  ax-cnex 7933
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-pnf 8025  df-mnf 8026  df-xr 8027  df-xneg 9804
This theorem is referenced by:  xnegcl  9864  xnegneg  9865  xltnegi  9867  xnegid  9891  xnegdi  9900  xsubge0  9913  xposdif  9914
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