Theorem elxnn0 9214

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 9213	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2242	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3274	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 7985	. . . 4 |- +oo e. _V
5	4	elsn2 3623	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 762	. 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 708   = wceq 1353   e. wcel 2146   u. cun 3125   {csn 3589   +oocpnf 7963   NN0cn0 9149  NN0*cxnn0 9212

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-un 4427  ax-cnex 7877

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806  df-pnf 7968  df-xr 7970  df-xnn0 9213

This theorem is referenced by:  xnn0xr  9217  pnf0xnn0  9219  xnn0nemnf  9223  xnn0nnn0pnf  9225  xnn0dcle  9773  xnn0letri  9774  xnn0lenn0nn0  9836  xnn0xadd0  9838