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Theorem xnegpnf 10467
Description: Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
Assertion
Ref Expression
xnegpnf  |-  - e  +oo  =  -oo

Proof of Theorem xnegpnf
StepHypRef Expression
1 df-xneg 10384 . 2  |-  - e  +oo  =  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo , 
+oo ,  -u  +oo )
)
2 eqid 2256 . . 3  |-  +oo  =  +oo
3 iftrue 3512 . . 3  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u  +oo ) )  =  -oo )
42, 3ax-mp 10 . 2  |-  if ( 
+oo  =  +oo ,  -oo ,  if (  +oo  =  -oo ,  +oo ,  -u 
+oo ) )  = 
-oo
51, 4eqtri 2276 1  |-  - e  +oo  =  -oo
Colors of variables: wff set class
Syntax hints:    = wceq 1619   ifcif 3506    +oocpnf 8797    -oocmnf 8798   -ucneg 8971    - ecxne 10381
This theorem is referenced by:  xnegcl  10471  xnegneg  10472  xltnegi  10474  xnegid  10494  xnegdi  10499  xaddass2  10501  xsubge0  10512  xlesubadd  10514  xmulneg1  10520  xmulmnf1  10527  xadddi2  10548  xrsdsreclblem  16344  xblss2  17885
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-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
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-if 3507  df-xneg 10384
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