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Theorem xnn0xr 8711
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
StepHypRef Expression
1 elxnn0 8708 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8652 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32rexrd 7516 . . 3  |-  ( A  e.  NN0  ->  A  e. 
RR* )
4 pnfxr 7519 . . . 4  |- +oo  e.  RR*
5 eleq1 2150 . . . 4  |-  ( A  = +oo  ->  ( A  e.  RR*  <-> +oo  e.  RR* ) )
64, 5mpbiri 166 . . 3  |-  ( A  = +oo  ->  A  e.  RR* )
73, 6jaoi 671 . 2  |-  ( ( A  e.  NN0  \/  A  = +oo )  ->  A  e.  RR* )
81, 7sylbi 119 1  |-  ( A  e. NN0*  ->  A  e.  RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664    = wceq 1289    e. wcel 1438   +oocpnf 7498   RR*cxr 7500   NN0cn0 8643  NN0*cxnn0 8706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-un 4251  ax-cnex 7415  ax-resscn 7416  ax-1re 7418  ax-addrcl 7421  ax-rnegex 7433
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-pnf 7503  df-xr 7505  df-inn 8395  df-n0 8644  df-xnn0 8707
This theorem is referenced by:  xnn0xrnemnf  8718
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