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Theorem xnegmnf 10752
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 10666 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10673 . . . 4  |-  +oo  =/=  -oo
32necomi 2649 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3707 . . 3  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u  -oo ) )  =  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
53, 4ax-mp 8 . 2  |-  if ( 
-oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u 
-oo ) )  =  if (  -oo  =  -oo ,  +oo ,  -u  -oo )
6 eqid 2404 . . 3  |-  -oo  =  -oo
7 iftrue 3705 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  +oo , 
-u  -oo )  =  +oo )
86, 7ax-mp 8 . 2  |-  if ( 
-oo  =  -oo ,  +oo ,  -u  -oo )  = 
+oo
91, 5, 83eqtri 2428 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1649    =/= wne 2567   ifcif 3699    +oocpnf 9073    -oocmnf 9074   -ucneg 9248    - ecxne 10663
This theorem is referenced by:  xnegcl  10755  xnegneg  10756  xltnegi  10758  xnegid  10778  xnegdi  10783  xsubge0  10796  xmulneg1  10804  xmulpnf1n  10813  xadddi2  10832  xrsdsreclblem  16699  xaddeq0  24072  xrge0npcan  24169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-pow 4337  ax-un 4660  ax-cnex 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-rex 2672  df-rab 2675  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-uni 3976  df-pnf 9078  df-mnf 9079  df-xr 9080  df-xneg 10666
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