Theorem elxnn0 9305

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 9304	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2260	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3300	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8073	. . . 4 |- +oo e. _V
5	4	elsn2 3652	. . 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 2164   u. cun 3151   {csn 3618   +oocpnf 8051   NN0cn0 9240  NN0*cxnn0 9303

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-un 4464  ax-cnex 7963

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-pnf 8056  df-xr 8058  df-xnn0 9304

This theorem is referenced by:  xnn0xr  9308  pnf0xnn0  9310  xnn0nemnf  9314  xnn0nnn0pnf  9316  xnn0dcle  9868  xnn0letri  9869  xnn0lenn0nn0  9931  xnn0xadd0  9933