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Theorem elrp 9134
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 3849 . 2  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
2 df-rp 9133 . 2  |-  RR+  =  { x  e.  RR  |  0  <  x }
31, 2elrab2 2774 1  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   class class class wbr 3845   RRcr 7347   0cc0 7348    < clt 7520   RR+crp 9132
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-rp 9133
This theorem is referenced by:  elrpii  9135  nnrp  9141  rpgt0  9143  rpregt0  9145  ralrp  9153  rexrp  9154  rpaddcl  9155  rpmulcl  9156  rpdivcl  9157  rpgecl  9160  rphalflt  9161  ge0p1rp  9163  rpnegap  9164  ltsubrp  9166  ltaddrp  9167  difrp  9168  elrpd  9169  iccdil  9413  icccntr  9415  expgt0  9984  sqrtdiv  10471  mulcn2  10697  ef01bndlem  11043
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