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Theorem xnegmnf 10531
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 10447 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10454 . . . 4  |-  +oo  =/=  -oo
32necomi 2529 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3574 . . 3  |-  (  -oo  =/=  +oo  ->  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u  -oo ) )  =  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
53, 4ax-mp 10 . 2  |-  if ( 
-oo  =  +oo ,  -oo ,  if (  -oo  =  -oo ,  +oo ,  -u 
-oo ) )  =  if (  -oo  =  -oo ,  +oo ,  -u  -oo )
6 eqid 2284 . . 3  |-  -oo  =  -oo
7 iftrue 3572 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  +oo , 
-u  -oo )  =  +oo )
86, 7ax-mp 10 . 2  |-  if ( 
-oo  =  -oo ,  +oo ,  -u  -oo )  = 
+oo
91, 5, 83eqtri 2308 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1624    =/= wne 2447   ifcif 3566    +oocpnf 8859    -oocmnf 8860   -ucneg 9033    - ecxne 10444
This theorem is referenced by:  xnegcl  10534  xnegneg  10535  xltnegi  10537  xnegid  10557  xnegdi  10562  xsubge0  10575  xmulneg1  10583  xmulpnf1n  10592  xadddi2  10611  xrsdsreclblem  16411
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-pow 4187  ax-un 4511  ax-cnex 8788
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-rex 2550  df-rab 2553  df-v 2791  df-un 3158  df-in 3160  df-ss 3167  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-uni 3829  df-pnf 8864  df-mnf 8865  df-xr 8866  df-xneg 10447
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