Theorem pnf0xnn0 9313

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 2193	. . 3 |- +oo = +oo
2	1	olci 733	. 2 |- ( +oo e. NN0 \/ +oo = +oo )
3		elxnn0 9308	. 2 |- ( +oo e. NN0* <-> ( +oo e. NN0 \/ +oo = +oo ) )
4	2, 3	mpbir 146	1 |- +oo e. NN0*

Syntax hints:   \/ wo 709   = wceq 1364   e. wcel 2164   +oocpnf 8053   NN0cn0 9243  NN0*cxnn0 9306

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-un 4465  ax-cnex 7965

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-pnf 8058  df-xr 8060  df-xnn0 9307

This theorem is referenced by:  inftonninf  10516  nninfctlemfo  12180  pcxnn0cl  12451