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Theorem xnegmnf 11532
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 11438 . 2  |-  -e -oo  =  if ( -oo  = +oo , -oo ,  if ( -oo  = -oo , +oo ,  -u -oo ) )
2 mnfnepnf 11447 . . 3  |- -oo  =/= +oo
3 ifnefalse 3905 . . 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 2462 . . 3  |- -oo  = -oo
65iftruei 3900 . 2  |-  if ( -oo  = -oo , +oo ,  -u -oo )  = +oo
71, 4, 63eqtri 2488 1  |-  -e -oo  = +oo
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    =/= wne 2633   ifcif 3893   +oocpnf 9698   -oocmnf 9699   -ucneg 9887    -ecxne 11435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-pow 4595  ax-un 6610  ax-cnex 9621
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-rex 2755  df-rab 2758  df-v 3059  df-un 3421  df-in 3423  df-ss 3430  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-uni 4213  df-pnf 9703  df-mnf 9704  df-xr 9705  df-xneg 11438
This theorem is referenced by:  xnegcl  11535  xnegneg  11536  xltnegi  11538  xnegid  11558  xnegdi  11563  xsubge0  11576  xmulneg1  11584  xmulpnf1n  11593  xadddi2  11612  xrsdsreclblem  19063  xaddeq0  28379  xrge0npcan  28506  carsgclctunlem2  29200
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