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Theorem xnegmnf 9779
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 9722 . 2  |-  -e -oo  =  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )
2 mnfnepnf 7968 . . 3  |- -oo  =/= +oo
3 ifnefalse 3536 . . 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 2170 . . 3  |- -oo  = -oo
65iftruei 3531 . 2  |-  if ( -oo  = -oo , +oo ,  -u -oo )  = +oo
71, 4, 63eqtri 2195 1  |-  -e -oo  = +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1348    =/= wne 2340   ifcif 3525   +oocpnf 7944   -oocmnf 7945   -ucneg 8084    -ecxne 9719
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-un 4416  ax-cnex 7858
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-pnf 7949  df-mnf 7950  df-xr 7951  df-xneg 9722
This theorem is referenced by:  xnegcl  9782  xnegneg  9783  xltnegi  9785  xnegid  9809  xnegdi  9818  xsubge0  9831  xposdif  9832
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