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Theorem rpre 8873
Description: A positive real is a real. (Contributed by NM, 27-Oct-2007.)
Assertion
Ref Expression
rpre  |-  ( A  e.  RR+  ->  A  e.  RR )

Proof of Theorem rpre
StepHypRef Expression
1 df-rp 8868 . . 3  |-  RR+  =  { x  e.  RR  |  0  <  x }
2 ssrab2 3088 . . 3  |-  { x  e.  RR  |  0  < 
x }  C_  RR
31, 2eqsstri 3038 . 2  |-  RR+  C_  RR
43sseli 3004 1  |-  ( A  e.  RR+  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   {crab 2357   class class class wbr 3805   RRcr 7094   0cc0 7095    < clt 7267   RR+crp 8867
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-in 2988  df-ss 2995  df-rp 8868
This theorem is referenced by:  rpxr  8874  rpcn  8875  rpssre  8877  rpge0  8879  rprege0  8881  rpap0  8883  rprene0  8884  rpreap0  8885  rpaddcl  8890  rpmulcl  8891  rpdivcl  8892  rpgecl  8895  ledivge1le  8936  addlelt  8972  iccdil  9148  expnlbnd  9746  caucvgre  10068  rennim  10089  rpsqrtcl  10128  qdenre  10289  2clim  10341  cn1lem  10353  climsqz  10374  climsqz2  10375  climcau  10385
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