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Theorem pnfnemnf 11440
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 11435 . . . 4  |- +oo  e.  RR*
2 pwne 4567 . . . 4  |-  ( +oo  e.  RR*  ->  ~P +oo  =/= +oo )
31, 2ax-mp 5 . . 3  |-  ~P +oo  =/= +oo
43necomi 2697 . 2  |- +oo  =/=  ~P +oo
5 df-mnf 9696 . 2  |- -oo  =  ~P +oo
64, 5neeqtrri 2716 1  |- +oo  =/= -oo
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1904    =/= wne 2641   ~Pcpw 3942   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-pow 4579  ax-un 6602  ax-cnex 9613
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-rex 2762  df-rab 2765  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-pw 3944  df-sn 3960  df-pr 3962  df-uni 4191  df-pnf 9695  df-mnf 9696  df-xr 9697
This theorem is referenced by:  mnfnepnf  11441  xrnemnf  11442  xrltnr  11444  pnfnlt  11453  nltmnf  11454  xaddpnf1  11542  xaddnemnf  11551  xmullem2  11576  xadddilem  11605  hashnemnf  12565  xrge0iifhom  28817  esumpr2  28962  xnn0nemnf  39232
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