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Theorem elrpd 9066
Description: Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
Hypotheses
Ref Expression
elrpd.1  |-  ( ph  ->  A  e.  RR )
elrpd.2  |-  ( ph  ->  0  <  A )
Assertion
Ref Expression
elrpd  |-  ( ph  ->  A  e.  RR+ )

Proof of Theorem elrpd
StepHypRef Expression
1 elrpd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 elrpd.2 . 2  |-  ( ph  ->  0  <  A )
3 elrp 9031 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
41, 2, 3sylanbrc 408 1  |-  ( ph  ->  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   class class class wbr 3811   RRcr 7252   0cc0 7253    < clt 7425   RR+crp 9029
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-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-rp 9030
This theorem is referenced by:  zltaddlt1le  9318  modqval  9620  ltexp2a  9844  leexp2a  9845  expnlbnd2  9914  resqrexlem1arp  10265  resqrexlemp1rp  10266  resqrexlemcalc2  10275  resqrexlemcalc3  10276  resqrexlemgt0  10280  resqrexlemglsq  10282  rpsqrtcl  10301  absrpclap  10321  mulcn2  10525  climge0  10537
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