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Theorem xnn0xr 9052
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 9049 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8993 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32rexrd 7822 . . 3  |-  ( A  e.  NN0  ->  A  e. 
RR* )
4 pnfxr 7825 . . . 4  |- +oo  e.  RR*
5 eleq1 2202 . . . 4  |-  ( A  = +oo  ->  ( A  e.  RR*  <-> +oo  e.  RR* ) )
64, 5mpbiri 167 . . 3  |-  ( A  = +oo  ->  A  e.  RR* )
73, 6jaoi 705 . 2  |-  ( ( A  e.  NN0  \/  A  = +oo )  ->  A  e.  RR* )
81, 7sylbi 120 1  |-  ( A  e. NN0*  ->  A  e.  RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 697    = wceq 1331    e. wcel 1480   +oocpnf 7804   RR*cxr 7806   NN0cn0 8984  NN0*cxnn0 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724  ax-rnegex 7736
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-pnf 7809  df-xr 7811  df-inn 8728  df-n0 8985  df-xnn0 9048
This theorem is referenced by:  xnn0xrnemnf  9059
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