Theorem xnn0xr 9244

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 9241	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9185	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 8007	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 8010	. . . 4 |- +oo e. RR*
5		eleq1 2240	. . . 4 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = +oo -> A e. RR* )
7	3, 6	jaoi 716	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A e. RR* )
8	1, 7	sylbi 121	1 |- ( A e. NN0* -> A e. RR* )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   e. wcel 2148   +oocpnf 7989   RR*cxr 7991   NN0cn0 9176  NN0*cxnn0 9239

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-rnegex 7920

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-pnf 7994  df-xr 7996  df-inn 8920  df-n0 9177  df-xnn0 9240

This theorem is referenced by:  xnn0xrnemnf  9251  xnn0dcle  9802  xnn0letri  9803