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Theorem pnfnemnf 11329
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 11324 . . . 4  |- +oo  e.  RR*
2 pwne 4603 . . . 4  |-  ( +oo  e.  RR*  ->  ~P +oo  =/= +oo )
31, 2ax-mp 5 . . 3  |-  ~P +oo  =/= +oo
43necomi 2724 . 2  |- +oo  =/=  ~P +oo
5 df-mnf 9620 . 2  |- -oo  =  ~P +oo
64, 5neeqtrri 2753 1  |- +oo  =/= -oo
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823    =/= wne 2649   ~Pcpw 3999   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-pow 4615  ax-un 6565  ax-cnex 9537
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-rex 2810  df-rab 2813  df-v 3108  df-un 3466  df-in 3468  df-ss 3475  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236  df-pnf 9619  df-mnf 9620  df-xr 9621
This theorem is referenced by:  mnfnepnf  11330  xrnemnf  11331  xrltnr  11333  pnfnlt  11340  nltmnf  11341  xaddpnf1  11428  xaddnemnf  11436  xmullem2  11460  xadddilem  11489  hashnemnf  12402  xrge0iifhom  28157  esumpr2  28299
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