Theorem xnn0xr 9308

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 9305	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9249	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 8069	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 8072	. . . 4 |- +oo e. RR*
5		eleq1 2256	. . . 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 1364   e. wcel 2164   +oocpnf 8051   RR*cxr 8053   NN0cn0 9240  NN0*cxnn0 9303

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 4147  ax-pow 4203  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-rnegex 7981

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-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-pnf 8056  df-xr 8058  df-inn 8983  df-n0 9241  df-xnn0 9304

This theorem is referenced by:  xnn0xrnemnf  9315  xnn0dcle  9868  xnn0letri  9869