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Theorem xnegpnf 9774
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 9718 . 2  |-  -e +oo  =  if ( +oo  = +oo , -oo ,  if ( +oo  = -oo , +oo ,  -u +oo ) )
2 eqid 2170 . . 3  |- +oo  = +oo
32iftruei 3531 . 2  |-  if ( +oo  = +oo , -oo ,  if ( +oo  = -oo , +oo ,  -u +oo ) )  = -oo
41, 3eqtri 2191 1  |-  -e +oo  = -oo
Colors of variables: wff set class
Syntax hints:    = wceq 1348   ifcif 3525   +oocpnf 7940   -oocmnf 7941   -ucneg 8080    -ecxne 9715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-if 3526  df-xneg 9718
This theorem is referenced by:  xnegcl  9778  xnegneg  9779  xltnegi  9781  xnegid  9805  xnegdi  9814  xaddass2  9816  xsubge0  9827  xposdif  9828  xlesubadd  9829  xblss2ps  13159  xblss2  13160
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