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Theorem xnegmnf 9831
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 9774 . 2  |-  -e -oo  =  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )
2 mnfnepnf 8015 . . 3  |- -oo  =/= +oo
3 ifnefalse 3547 . . 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 2177 . . 3  |- -oo  = -oo
65iftruei 3542 . 2  |-  if ( -oo  = -oo , +oo ,  -u -oo )  = +oo
71, 4, 63eqtri 2202 1  |-  -e -oo  = +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1353    =/= wne 2347   ifcif 3536   +oocpnf 7991   -oocmnf 7992   -ucneg 8131    -ecxne 9771
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435  ax-cnex 7904
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-pnf 7996  df-mnf 7997  df-xr 7998  df-xneg 9774
This theorem is referenced by:  xnegcl  9834  xnegneg  9835  xltnegi  9837  xnegid  9861  xnegdi  9870  xsubge0  9883  xposdif  9884
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