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Theorem xnn0xr 9013
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 9010 . 2  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
2 nn0re 8954 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
32rexrd 7783 . . 3  |-  ( A  e.  NN0  ->  A  e. 
RR* )
4 pnfxr 7786 . . . 4  |- +oo  e.  RR*
5 eleq1 2180 . . . 4  |-  ( A  = +oo  ->  ( A  e.  RR*  <-> +oo  e.  RR* ) )
64, 5mpbiri 167 . . 3  |-  ( A  = +oo  ->  A  e.  RR* )
73, 6jaoi 690 . 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 682    = wceq 1316    e. wcel 1465   +oocpnf 7765   RR*cxr 7767   NN0cn0 8945  NN0*cxnn0 9008
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-rnegex 7697
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-pnf 7770  df-xr 7772  df-inn 8689  df-n0 8946  df-xnn0 9009
This theorem is referenced by:  xnn0xrnemnf  9020
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