Theorem elxnn0 9360

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 9359	. . 3 |- NN0* = ( NN0 u. { +oo } )
2	1	eleq2i 2272	. 2 |- ( A e. NN0* <-> A e. ( NN0 u. { +oo } ) )
3		elun 3314	. 2 |- ( A e. ( NN0 u. { +oo } ) <-> ( A e. NN0 \/ A e. { +oo } ) )
4		pnfex 8126	. . . 4 |- +oo e. _V
5	4	elsn2 3667	. . 3 |- ( A e. { +oo } <-> A = +oo )
6	5	orbi2i 764	. 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 710   = wceq 1373   e. wcel 2176   u. cun 3164   {csn 3633   +oocpnf 8104   NN0cn0 9295  NN0*cxnn0 9358

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-un 4480  ax-cnex 8016

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-pnf 8109  df-xr 8111  df-xnn0 9359

This theorem is referenced by:  xnn0xr  9363  pnf0xnn0  9365  xnn0nemnf  9369  xnn0nnn0pnf  9371  xnn0dcle  9924  xnn0letri  9925  xnn0lenn0nn0  9987  xnn0xadd0  9989