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Theorem xnn0nemnf 8641
Description: No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0nemnf  |-  ( A  e. NN0*  ->  A  =/= -oo )

Proof of Theorem xnn0nemnf
StepHypRef Expression
1 elxnn0 8632 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8574 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32renemnfd 7442 . . 3  |-  ( A  e.  NN0  ->  A  =/= -oo )
4 pnfnemnf 7445 . . . 4  |- +oo  =/= -oo
5 neeq1 2262 . . . 4  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
64, 5mpbiri 166 . . 3  |-  ( A  = +oo  ->  A  =/= -oo )
73, 6jaoi 669 . 2  |-  ( ( A  e.  NN0  \/  A  = +oo )  ->  A  =/= -oo )
81, 7sylbi 119 1  |-  ( A  e. NN0*  ->  A  =/= -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285    e. wcel 1434    =/= wne 2249   +oocpnf 7422   -oocmnf 7423   NN0cn0 8565  NN0*cxnn0 8628
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1re 7342  ax-addrcl 7345  ax-rnegex 7357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-pnf 7427  df-mnf 7428  df-xr 7429  df-inn 8317  df-n0 8566  df-xnn0 8630
This theorem is referenced by:  xnn0xrnemnf  8642
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