Theorem elxnn0 9066

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 9065	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2207	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3222	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 7843	. . . 4 |- +oo e. _V
5	4	elsn2 3566	. . 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 1332   e. wcel 1481   u. cun 3074   {csn 3532   +oocpnf 7821   NN0cn0 9001  NN0*cxnn0 9064

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-un 4363  ax-cnex 7735

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-pnf 7826  df-xr 7828  df-xnn0 9065

This theorem is referenced by:  xnn0xr  9069  pnf0xnn0  9071  xnn0nemnf  9075  xnn0nnn0pnf  9077  xnn0lenn0nn0  9678  xnn0xadd0  9680