Theorem elxnn0 8632

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 8630	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2149	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3125	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 7444	. . . 4 |- +oo e. _V
5	4	elsn2 3452	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 712	. 2 |- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2, 3, 6	3bitri 204	1 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Syntax hints:   <-> wb 103   \/ wo 662   = wceq 1285   e. wcel 1434   u. cun 2982   {csn 3422   +oocpnf 7422   NN0cn0 8565  NN0*cxnn0 8628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-un 4224  ax-cnex 7339

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-pnf 7427  df-xr 7429  df-xnn0 8630

This theorem is referenced by:  xnn0xr  8635  pnf0xnn0  8637  xnn0nemnf  8641  xnn0nnn0pnf  8643