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Theorem elxnn0 9272
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 9271 . . 3  |- NN0*  =  ( NN0  u.  { +oo } )
21eleq2i 2256 . 2  |-  ( A  e. NN0* 
<->  A  e.  ( NN0 
u.  { +oo } ) )
3 elun 3291 . 2  |-  ( A  e.  ( NN0  u.  { +oo } )  <->  ( A  e.  NN0  \/  A  e. 
{ +oo } ) )
4 pnfex 8042 . . . 4  |- +oo  e.  _V
54elsn2 3641 . . 3  |-  ( A  e.  { +oo }  <->  A  = +oo )
65orbi2i 763 . 2  |-  ( ( A  e.  NN0  \/  A  e.  { +oo }
)  <->  ( A  e. 
NN0  \/  A  = +oo ) )
72, 3, 63bitri 206 1  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160    u. cun 3142   {csn 3607   +oocpnf 8020   NN0cn0 9207  NN0*cxnn0 9270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-un 4451  ax-cnex 7933
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-pnf 8025  df-xr 8027  df-xnn0 9271
This theorem is referenced by:  xnn0xr  9275  pnf0xnn0  9277  xnn0nemnf  9281  xnn0nnn0pnf  9283  xnn0dcle  9834  xnn0letri  9835  xnn0lenn0nn0  9897  xnn0xadd0  9899
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