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Theorem elxnn0 9054
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 9053 . . 3  |- NN0*  =  ( NN0  u.  { +oo } )
21eleq2i 2206 . 2  |-  ( A  e. NN0* 
<->  A  e.  ( NN0 
u.  { +oo } ) )
3 elun 3217 . 2  |-  ( A  e.  ( NN0  u.  { +oo } )  <->  ( A  e.  NN0  \/  A  e. 
{ +oo } ) )
4 pnfex 7831 . . . 4  |- +oo  e.  _V
54elsn2 3559 . . 3  |-  ( A  e.  { +oo }  <->  A  = +oo )
65orbi2i 751 . 2  |-  ( ( A  e.  NN0  \/  A  e.  { +oo }
)  <->  ( A  e. 
NN0  \/  A  = +oo ) )
72, 3, 63bitri 205 1  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480    u. cun 3069   {csn 3527   +oocpnf 7809   NN0cn0 8989  NN0*cxnn0 9052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355  ax-cnex 7723
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7814  df-xr 7816  df-xnn0 9053
This theorem is referenced by:  xnn0xr  9057  pnf0xnn0  9059  xnn0nemnf  9063  xnn0nnn0pnf  9065  xnn0lenn0nn0  9660  xnn0xadd0  9662
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