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Theorem xnn0nemnf 8717
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 8708 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8652 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32renemnfd 7518 . . 3  |-  ( A  e.  NN0  ->  A  =/= -oo )
4 pnfnemnf 7521 . . . 4  |- +oo  =/= -oo
5 neeq1 2268 . . . 4  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
64, 5mpbiri 166 . . 3  |-  ( A  = +oo  ->  A  =/= -oo )
73, 6jaoi 671 . 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 664    = wceq 1289    e. wcel 1438    =/= wne 2255   +oocpnf 7498   -oocmnf 7499   NN0cn0 8643  NN0*cxnn0 8706
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1re 7418  ax-addrcl 7421  ax-rnegex 7433
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-pnf 7503  df-mnf 7504  df-xr 7505  df-inn 8395  df-n0 8644  df-xnn0 8707
This theorem is referenced by:  xnn0xrnemnf  8718
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