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Theorem elxnn0 8632
Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
elxnn0  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)

Proof of Theorem elxnn0
StepHypRef Expression
1 df-xnn0 8630 . . 3  |- NN0*  =  ( NN0  u.  { +oo } )
21eleq2i 2149 . 2  |-  ( A  e. NN0* 
<->  A  e.  ( NN0 
u.  { +oo } ) )
3 elun 3125 . 2  |-  ( A  e.  ( NN0  u.  { +oo } )  <->  ( A  e.  NN0  \/  A  e. 
{ +oo } ) )
4 pnfex 7444 . . . 4  |- +oo  e.  _V
54elsn2 3452 . . 3  |-  ( A  e.  { +oo }  <->  A  = +oo )
65orbi2i 712 . 2  |-  ( ( A  e.  NN0  \/  A  e.  { +oo }
)  <->  ( A  e. 
NN0  \/  A  = +oo ) )
72, 3, 63bitri 204 1  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434    u. cun 2982   {csn 3422   +oocpnf 7422   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-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-cnex 7339
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-pnf 7427  df-xr 7429  df-xnn0 8630
This theorem is referenced by:  xnn0xr  8635  pnf0xnn0  8637  xnn0nemnf  8641  xnn0nnn0pnf  8643
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