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Theorem elrp 9591
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 3986 . 2  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
2 df-rp 9590 . 2  |-  RR+  =  { x  e.  RR  |  0  <  x }
31, 2elrab2 2885 1  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 2136   class class class wbr 3982   RRcr 7752   0cc0 7753    < clt 7933   RR+crp 9589
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-rp 9590
This theorem is referenced by:  elrpii  9592  nnrp  9599  rpgt0  9601  rpregt0  9603  ralrp  9611  rexrp  9612  rpaddcl  9613  rpmulcl  9614  rpdivcl  9615  rpgecl  9618  rphalflt  9619  ge0p1rp  9621  rpnegap  9622  negelrp  9623  ltsubrp  9626  ltaddrp  9627  difrp  9628  elrpd  9629  iccdil  9934  icccntr  9936  dfrp2  10199  expgt0  10488  sqrtdiv  10984  mulcn2  11253  ef01bndlem  11697  nconstwlpolem  13943
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