Theorem elxnn0 9200

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 9199	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2237	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3268	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 7973	. . . 4 |- +oo e. _V
5	4	elsn2 3617	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 757	. 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 703   = wceq 1348   e. wcel 2141   u. cun 3119   {csn 3583   +oocpnf 7951   NN0cn0 9135  NN0*cxnn0 9198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418  ax-cnex 7865

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-pnf 7956  df-xr 7958  df-xnn0 9199

This theorem is referenced by:  xnn0xr  9203  pnf0xnn0  9205  xnn0nemnf  9209  xnn0nnn0pnf  9211  xnn0dcle  9759  xnn0letri  9760  xnn0lenn0nn0  9822  xnn0xadd0  9824