Theorem xnn0xr 9362

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 9359	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9303	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 8121	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 8124	. . . 4 |- +oo e. RR*
5		eleq1 2267	. . . 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 1372   e. wcel 2175   +oocpnf 8103   RR*cxr 8105   NN0cn0 9294  NN0*cxnn0 9357

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021  ax-rnegex 8033

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-pnf 8108  df-xr 8110  df-inn 9036  df-n0 9295  df-xnn0 9358

This theorem is referenced by:  xnn0xrnemnf  9369  xnn0dcle  9923  xnn0letri  9924