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Theorem pnfnemnf 8015
Description: Plus and minus infinity are different elements of  RR*. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
pnfnemnf  |- +oo  =/= -oo

Proof of Theorem pnfnemnf
StepHypRef Expression
1 pnfxr 8013 . . . 4  |- +oo  e.  RR*
2 pwne 4162 . . . 4  |-  ( +oo  e.  RR*  ->  ~P +oo  =/= +oo )
31, 2ax-mp 5 . . 3  |-  ~P +oo  =/= +oo
43necomi 2432 . 2  |- +oo  =/=  ~P +oo
5 df-mnf 7998 . 2  |- -oo  =  ~P +oo
64, 5neeqtrri 2376 1  |- +oo  =/= -oo
Colors of variables: wff set class
Syntax hints:    e. wcel 2148    =/= wne 2347   ~Pcpw 3577   +oocpnf 7992   -oocmnf 7993   RR*cxr 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435  ax-cnex 7905
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-pnf 7997  df-mnf 7998  df-xr 7999
This theorem is referenced by:  mnfnepnf  8016  xnn0nemnf  9253  xrnemnf  9780  xrltnr  9782  pnfnlt  9790  nltmnf  9791  ngtmnft  9820  xrmnfdc  9846  xaddpnf1  9849  xaddnemnf  9860  xposdif  9885  xleaddadd  9890
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