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Theorem elxnn0 8671
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 8670 . . 3  |- NN0*  =  ( NN0  u.  { +oo } )
21eleq2i 2151 . 2  |-  ( A  e. NN0* 
<->  A  e.  ( NN0 
u.  { +oo } ) )
3 elun 3130 . 2  |-  ( A  e.  ( NN0  u.  { +oo } )  <->  ( A  e.  NN0  \/  A  e. 
{ +oo } ) )
4 pnfex 7485 . . . 4  |- +oo  e.  _V
54elsn2 3461 . . 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 1287    e. wcel 1436    u. cun 2986   {csn 3431   +oocpnf 7463   NN0cn0 8606  NN0*cxnn0 8669
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 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 3932  ax-pow 3984  ax-un 4234  ax-cnex 7380
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-pnf 7468  df-xr 7470  df-xnn0 8670
This theorem is referenced by:  xnn0xr  8674  pnf0xnn0  8676  xnn0nemnf  8680  xnn0nnn0pnf  8682
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