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Theorem xnegmnf 10785
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 10699 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10706 . . . 4  |-  +oo  =/=  -oo
32necomi 2680 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3739 . . 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 2435 . . 3  |-  -oo  =  -oo
7 iftrue 3737 . . 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 2459 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1652    =/= wne 2598   ifcif 3731    +oocpnf 9106    -oocmnf 9107   -ucneg 9281    - ecxne 10696
This theorem is referenced by:  xnegcl  10788  xnegneg  10789  xltnegi  10791  xnegid  10811  xnegdi  10816  xsubge0  10829  xmulneg1  10837  xmulpnf1n  10846  xadddi2  10865  xrsdsreclblem  16732  xaddeq0  24107  xrge0npcan  24204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-pow 4369  ax-un 4692  ax-cnex 9035
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-rex 2703  df-rab 2706  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-uni 4008  df-pnf 9111  df-mnf 9112  df-xr 9113  df-xneg 10699
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