Theorem elxnn0 11365

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 11364	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2693	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3753	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 10093	. . . 4 |- +oo e. _V
5	4	elsn2 4211	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 541	. 2 |- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2, 3, 6	3bitri 286	1 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Syntax hints:   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   +oocpnf 10071   NN0cn0 11292  NN0*cxnn0 11363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-pnf 10076  df-xr 10078  df-xnn0 11364

This theorem is referenced by:  xnn0xr  11368  pnf0xnn0  11370  xnn0nemnf  11374  xnn0nnn0pnf  11376  xnn0n0n1ge2b  11965  xnn0ge0  11967  xnn0lenn0nn0  12075  xnn0xadd0  12077  xnn0xrge0  12325  tayl0  24116