Theorem elxnn0 9611

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 9610	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2305	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3370	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8369	. . . 4 |- +oo e. _V
5	4	elsn2 3739	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 774	. 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 720   = wceq 1402   e. wcel 2209   u. cun 3218   {csn 3705   +oocpnf 8347   NN0cn0 9542  NN0*cxnn0 9609

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-un 4573  ax-cnex 8260

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-pnf 8352  df-xr 8354  df-xnn0 9610

This theorem is referenced by:  xnn0xr  9614  pnf0xnn0  9616  xnn0nemnf  9620  xnn0nnn0pnf  9622  xnn0dcle  10183  xnn0letri  10184  xnn0lenn0nn0  10246  xnn0xadd0  10248