Theorem elxnn0 9179

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 9178	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2233	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3263	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 7952	. . . 4 |- +oo e. _V
5	4	elsn2 3610	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 752	. 2 |- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2, 3, 6	3bitri 205	1 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Syntax hints:   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   u. cun 3114   {csn 3576   +oocpnf 7930   NN0cn0 9114  NN0*cxnn0 9177

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-un 4411  ax-cnex 7844

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-pnf 7935  df-xr 7937  df-xnn0 9178

This theorem is referenced by:  xnn0xr  9182  pnf0xnn0  9184  xnn0nemnf  9188  xnn0nnn0pnf  9190  xnn0dcle  9738  xnn0letri  9739  xnn0lenn0nn0  9801  xnn0xadd0  9803