Theorem elxnn0 9434

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 9433	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2296	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3345	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8200	. . . 4 |- +oo e. _V
5	4	elsn2 3700	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 767	. 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 713   = wceq 1395   e. wcel 2200   u. cun 3195   {csn 3666   +oocpnf 8178   NN0cn0 9369  NN0*cxnn0 9432

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-un 4524  ax-cnex 8090

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-pnf 8183  df-xr 8185  df-xnn0 9433

This theorem is referenced by:  xnn0xr  9437  pnf0xnn0  9439  xnn0nemnf  9443  xnn0nnn0pnf  9445  xnn0dcle  9998  xnn0letri  9999  xnn0lenn0nn0  10061  xnn0xadd0  10063