Theorem elxnn0 9565

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 9564	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2299	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3360	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8327	. . . 4 |- +oo e. _V
5	4	elsn2 3723	. . 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 2203   u. cun 3209   {csn 3689   +oocpnf 8305   NN0cn0 9496  NN0*cxnn0 9563

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-un 4554  ax-cnex 8218

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-pnf 8310  df-xr 8312  df-xnn0 9564

This theorem is referenced by:  xnn0xr  9568  pnf0xnn0  9570  xnn0nemnf  9574  xnn0nnn0pnf  9576  xnn0dcle  10135  xnn0letri  10136  xnn0lenn0nn0  10198  xnn0xadd0  10200