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Theorem xnegmnf 9619
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 9566 . 2  |-  -e -oo  =  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )
2 mnfnepnf 7828 . . 3  |- -oo  =/= +oo
3 ifnefalse 3485 . . 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 2139 . . 3  |- -oo  = -oo
65iftruei 3480 . 2  |-  if ( -oo  = -oo , +oo ,  -u -oo )  = +oo
71, 4, 63eqtri 2164 1  |-  -e -oo  = +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1331    =/= wne 2308   ifcif 3474   +oocpnf 7804   -oocmnf 7805   -ucneg 7941    -ecxne 9563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355  ax-cnex 7718
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7809  df-mnf 7810  df-xr 7811  df-xneg 9566
This theorem is referenced by:  xnegcl  9622  xnegneg  9623  xltnegi  9625  xnegid  9649  xnegdi  9658  xsubge0  9671  xposdif  9672
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