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Theorem xnegmnf 10553
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 10468 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10475 . . . 4  |-  +oo  =/=  -oo
32necomi 2541 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3586 . . 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 2296 . . 3  |-  -oo  =  -oo
7 iftrue 3584 . . 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 2320 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1632    =/= wne 2459   ifcif 3578    +oocpnf 8880    -oocmnf 8881   -ucneg 9054    - ecxne 10465
This theorem is referenced by:  xnegcl  10556  xnegneg  10557  xltnegi  10559  xnegid  10579  xnegdi  10584  xsubge0  10597  xmulneg1  10605  xmulpnf1n  10614  xadddi2  10633  xrsdsreclblem  16433  xaddeq0  23319  xrge0npcan  23348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-pow 4204  ax-un 4528  ax-cnex 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844  df-pnf 8885  df-mnf 8886  df-xr 8887  df-xneg 10468
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