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Theorem elrp 9639
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 4004 . 2  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
2 df-rp 9638 . 2  |-  RR+  =  { x  e.  RR  |  0  <  x }
31, 2elrab2 2896 1  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   class class class wbr 4000   RRcr 7798   0cc0 7799    < clt 7979   RR+crp 9637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-rp 9638
This theorem is referenced by:  elrpii  9640  nnrp  9647  rpgt0  9649  rpregt0  9651  ralrp  9659  rexrp  9660  rpaddcl  9661  rpmulcl  9662  rpdivcl  9663  rpgecl  9666  rphalflt  9667  ge0p1rp  9669  rpnegap  9670  negelrp  9671  ltsubrp  9674  ltaddrp  9675  difrp  9676  elrpd  9677  iccdil  9982  icccntr  9984  dfrp2  10247  expgt0  10536  sqrtdiv  11032  mulcn2  11301  ef01bndlem  11745  nconstwlpolem  14461
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