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Theorem elrp 9411
Description: Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
Assertion
Ref Expression
elrp  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )

Proof of Theorem elrp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq2 3903 . 2  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
2 df-rp 9410 . 2  |-  RR+  =  { x  e.  RR  |  0  <  x }
31, 2elrab2 2816 1  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1465   class class class wbr 3899   RRcr 7587   0cc0 7588    < clt 7768   RR+crp 9409
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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-rp 9410
This theorem is referenced by:  elrpii  9412  nnrp  9419  rpgt0  9421  rpregt0  9423  ralrp  9431  rexrp  9432  rpaddcl  9433  rpmulcl  9434  rpdivcl  9435  rpgecl  9438  rphalflt  9439  ge0p1rp  9441  rpnegap  9442  negelrp  9443  ltsubrp  9446  ltaddrp  9447  difrp  9448  elrpd  9449  iccdil  9749  icccntr  9751  expgt0  10294  sqrtdiv  10782  mulcn2  11049  ef01bndlem  11390
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