Theorem xnn0xr 9069

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

Assertion

Ref	Expression
xnn0xr	|- ( A e. NN0* -> A e. RR* )

Proof of Theorem xnn0xr

Step	Hyp	Ref	Expression
1		elxnn0 9066	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9010	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 7839	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 7842	. . . 4 |- +oo e. RR*
5		eleq1 2203	. . . 4 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4, 5	mpbiri 167	. . 3 |- ( A = +oo -> A e. RR* )
7	3, 6	jaoi 706	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A e. RR* )
8	1, 7	sylbi 120	1 |- ( A e. NN0* -> A e. RR* )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 1481   +oocpnf 7821   RR*cxr 7823   NN0cn0 9001  NN0*cxnn0 9064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-un 4363  ax-cnex 7735  ax-resscn 7736  ax-1re 7738  ax-addrcl 7741  ax-rnegex 7753

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-pnf 7826  df-xr 7828  df-inn 8745  df-n0 9002  df-xnn0 9065

This theorem is referenced by:  xnn0xrnemnf  9076