Theorem xnn0xr 9257

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 9254	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9198	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 8020	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 8023	. . . 4 |- +oo e. RR*
5		eleq1 2250	. . . 4 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = +oo -> A e. RR* )
7	3, 6	jaoi 717	. 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 709   = wceq 1363   e. wcel 2158   +oocpnf 8002   RR*cxr 8004   NN0cn0 9189  NN0*cxnn0 9252

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921  ax-rnegex 7933

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-pnf 8007  df-xr 8009  df-inn 8933  df-n0 9190  df-xnn0 9253

This theorem is referenced by:  xnn0xrnemnf  9264  xnn0dcle  9815  xnn0letri  9816