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Theorem xnn0xr 8651
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 8648 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8592 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32rexrd 7458 . . 3  |-  ( A  e.  NN0  ->  A  e. 
RR* )
4 pnfxr 7461 . . . 4  |- +oo  e.  RR*
5 eleq1 2147 . . . 4  |-  ( A  = +oo  ->  ( A  e.  RR*  <-> +oo  e.  RR* ) )
64, 5mpbiri 166 . . 3  |-  ( A  = +oo  ->  A  e.  RR* )
73, 6jaoi 669 . 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 662    = wceq 1287    e. wcel 1436   +oocpnf 7440   RR*cxr 7442   NN0cn0 8583  NN0*cxnn0 8646
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-un 4227  ax-cnex 7357  ax-resscn 7358  ax-1re 7360  ax-addrcl 7363  ax-rnegex 7375
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-int 3666  df-pnf 7445  df-xr 7447  df-inn 8335  df-n0 8584  df-xnn0 8647
This theorem is referenced by:  xnn0xrnemnf  8658
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