Theorem elxnn0 9243

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

Step	Hyp	Ref	Expression
1		df-xnn0 9242	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2244	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3278	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8013	. . . 4 |- +oo e. _V
5	4	elsn2 3628	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 762	. 2 |- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2, 3, 6	3bitri 206	1 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Syntax hints:   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   u. cun 3129   {csn 3594   +oocpnf 7991   NN0cn0 9178  NN0*cxnn0 9241

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435  ax-cnex 7904

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-pnf 7996  df-xr 7998  df-xnn0 9242

This theorem is referenced by:  xnn0xr  9246  pnf0xnn0  9248  xnn0nemnf  9252  xnn0nnn0pnf  9254  xnn0dcle  9804  xnn0letri  9805  xnn0lenn0nn0  9867  xnn0xadd0  9869