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Theorem xnegmnf 10537
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 10452 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10459 . . . 4  |-  +oo  =/=  -oo
32necomi 2528 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3573 . . 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 2283 . . 3  |-  -oo  =  -oo
7 iftrue 3571 . . 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 2307 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1623    =/= wne 2446   ifcif 3565    +oocpnf 8864    -oocmnf 8865   -ucneg 9038    - ecxne 10449
This theorem is referenced by:  xnegcl  10540  xnegneg  10541  xltnegi  10543  xnegid  10563  xnegdi  10568  xsubge0  10581  xmulneg1  10589  xmulpnf1n  10598  xadddi2  10617  xrsdsreclblem  16417  xaddeq0  23304  xrge0npcan  23333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-pow 4188  ax-un 4512  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-rex 2549  df-rab 2552  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-pnf 8869  df-mnf 8870  df-xr 8871  df-xneg 10452
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