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Theorem pnfnemnf 10673
Description: Plus and minus infinity are distinguished elements of 
RR*. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
pnfnemnf  |-  +oo  =/=  -oo

Proof of Theorem pnfnemnf
StepHypRef Expression
1 pnfxr 10669 . . . 4  |-  +oo  e.  RR*
2 pwne 4326 . . . 4  |-  (  +oo  e.  RR*  ->  ~P  +oo  =/=  +oo )
31, 2ax-mp 8 . . 3  |-  ~P  +oo  =/=  +oo
43necomi 2649 . 2  |-  +oo  =/=  ~P 
+oo
5 df-mnf 9079 . 2  |-  -oo  =  ~P  +oo
64, 5neeqtrri 2590 1  |-  +oo  =/=  -oo
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    =/= wne 2567   ~Pcpw 3759    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075
This theorem is referenced by:  xrnemnf  10674  xrnepnf  10675  xrltnr  10676  pnfnlt  10681  nltmnf  10682  xnegmnf  10752  xaddpnf1  10768  xaddmnf1  10770  xaddmnf2  10771  mnfaddpnf  10773  xaddnemnf  10776  xaddnepnf  10777  xmullem2  10800  xadddilem  10829  resup  11203  hashnemnf  11583  xrge0iifhom  24276  esumpr2  24411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-pow 4337  ax-un 4660  ax-cnex 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-rex 2672  df-rab 2675  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-pw 3761  df-sn 3780  df-pr 3781  df-uni 3976  df-pnf 9078  df-mnf 9079  df-xr 9080
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