Theorem pnf0xnn0 8899

Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
pnf0xnn0	|- +oo e. NN0*

Proof of Theorem pnf0xnn0

Step	Hyp	Ref	Expression
1		eqid 2100	. . 3 |- +oo = +oo
2	1	olci 692	. 2 |- ( +oo e. NN0 \/ +oo = +oo )
3		elxnn0 8894	. 2 |- ( +oo e. NN0* <-> ( +oo e. NN0 \/ +oo = +oo ) )
4	2, 3	mpbir 145	1 |- +oo e. NN0*

Syntax hints:   \/ wo 670   = wceq 1299   e. wcel 1448   +oocpnf 7669   NN0cn0 8829  NN0*cxnn0 8892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-un 4293  ax-cnex 7586

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-pnf 7674  df-xr 7676  df-xnn0 8893

This theorem is referenced by:  inftonninf  10055