Theorem elxnn0 9272

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 9271	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2256	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3291	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8042	. . . 4 |- +oo e. _V
5	4	elsn2 3641	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 763	. 2 |- ( ( A e. NN0 \/ A e. { +oo } ) <-> ( A e. NN0 \/ A = +oo ) )
7	2, 3, 6	3bitri 206	1 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )

Syntax hints:   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   u. cun 3142   {csn 3607   +oocpnf 8020   NN0cn0 9207  NN0*cxnn0 9270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-un 4451  ax-cnex 7933

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-pnf 8025  df-xr 8027  df-xnn0 9271

This theorem is referenced by:  xnn0xr  9275  pnf0xnn0  9277  xnn0nemnf  9281  xnn0nnn0pnf  9283  xnn0dcle  9834  xnn0letri  9835  xnn0lenn0nn0  9897  xnn0xadd0  9899