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Theorem pnfnemnf 11325
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 11320 . . . 4  |- +oo  e.  RR*
2 pwne 4613 . . . 4  |-  ( +oo  e.  RR*  ->  ~P +oo  =/= +oo )
31, 2ax-mp 5 . . 3  |-  ~P +oo  =/= +oo
43necomi 2737 . 2  |- +oo  =/=  ~P +oo
5 df-mnf 9630 . 2  |- -oo  =  ~P +oo
64, 5neeqtrri 2766 1  |- +oo  =/= -oo
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767    =/= wne 2662   ~Pcpw 4010   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-pow 4625  ax-un 6575  ax-cnex 9547
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-rex 2820  df-rab 2823  df-v 3115  df-un 3481  df-in 3483  df-ss 3490  df-pw 4012  df-sn 4028  df-pr 4030  df-uni 4246  df-pnf 9629  df-mnf 9630  df-xr 9631
This theorem is referenced by:  mnfnepnf  11326  xrnemnf  11327  xrltnr  11329  pnfnlt  11336  nltmnf  11337  xaddpnf1  11424  xaddnemnf  11432  xmullem2  11456  xadddilem  11485  hashnemnf  12384  xrge0iifhom  27571  esumpr2  27730
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