Theorem elxnn0 9466

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 9465	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2298	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3348	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8232	. . . 4 |- +oo e. _V
5	4	elsn2 3703	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 769	. 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 715   = wceq 1397   e. wcel 2202   u. cun 3198   {csn 3669   +oocpnf 8210   NN0cn0 9401  NN0*cxnn0 9464

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-un 4530  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-pnf 8215  df-xr 8217  df-xnn0 9465

This theorem is referenced by:  xnn0xr  9469  pnf0xnn0  9471  xnn0nemnf  9475  xnn0nnn0pnf  9477  xnn0dcle  10036  xnn0letri  10037  xnn0lenn0nn0  10099  xnn0xadd0  10101