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Theorem xnegmnf 10489
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 10405 . 2  |-  - e  -oo  =  if (  -oo  =  +oo ,  -oo ,  if (  -oo  =  -oo , 
+oo ,  -u  -oo )
)
2 pnfnemnf 10412 . . . 4  |-  +oo  =/=  -oo
32necomi 2501 . . 3  |-  -oo  =/=  +oo
4 ifnefalse 3533 . . 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 2256 . . 3  |-  -oo  =  -oo
7 iftrue 3531 . . 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 2280 1  |-  - e  -oo  =  +oo
Colors of variables: wff set class
Syntax hints:    = wceq 1619    =/= wne 2419   ifcif 3525    +oocpnf 8818    -oocmnf 8819   -ucneg 8992    - ecxne 10402
This theorem is referenced by:  xnegcl  10492  xnegneg  10493  xltnegi  10495  xnegid  10515  xnegdi  10520  xsubge0  10533  xmulneg1  10541  xmulpnf1n  10550  xadddi2  10569  xrsdsreclblem  16365
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-pow 4146  ax-un 4470  ax-cnex 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-rex 2522  df-rab 2525  df-v 2759  df-un 3118  df-in 3120  df-ss 3127  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-uni 3788  df-pnf 8823  df-mnf 8824  df-xr 8825  df-xneg 10405
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