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Theorem pnfnemnf 7463
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 7461 . . . 4  |- +oo  e.  RR*
2 pwne 3963 . . . 4  |-  ( +oo  e.  RR*  ->  ~P +oo  =/= +oo )
31, 2ax-mp 7 . . 3  |-  ~P +oo  =/= +oo
43necomi 2336 . 2  |- +oo  =/=  ~P +oo
5 df-mnf 7446 . 2  |- -oo  =  ~P +oo
64, 5neeqtrri 2280 1  |- +oo  =/= -oo
Colors of variables: wff set class
Syntax hints:    e. wcel 1436    =/= wne 2251   ~Pcpw 3409   +oocpnf 7440   -oocmnf 7441   RR*cxr 7442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-un 4227  ax-cnex 7357
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-rex 2361  df-rab 2364  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-pnf 7445  df-mnf 7446  df-xr 7447
This theorem is referenced by:  mnfnepnf  7464  xnn0nemnf  8657  xrnemnf  9157  xrltnr  9159  pnfnlt  9166  nltmnf  9167  ngtmnft  9189
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