Theorem elxnn0 9582

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 9581	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2301	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3364	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8343	. . . 4 |- +oo e. _V
5	4	elsn2 3728	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 770	. 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 716   = wceq 1398   e. wcel 2205   u. cun 3212   {csn 3694   +oocpnf 8321   NN0cn0 9513  NN0*cxnn0 9580

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-un 4559  ax-cnex 8234

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-pnf 8326  df-xr 8328  df-xnn0 9581

This theorem is referenced by:  xnn0xr  9585  pnf0xnn0  9587  xnn0nemnf  9591  xnn0nnn0pnf  9593  xnn0dcle  10154  xnn0letri  10155  xnn0lenn0nn0  10217  xnn0xadd0  10219