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Theorem elrpd 9380
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 9345 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
41, 2, 3sylanbrc 411 1  |-  ( ph  ->  A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   class class class wbr 3895   RRcr 7546   0cc0 7547    < clt 7724   RR+crp 9343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rab 2399  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-rp 9344
This theorem is referenced by:  zltaddlt1le  9682  modqval  9990  ltexp2a  10238  leexp2a  10239  expnlbnd2  10310  resqrexlem1arp  10669  resqrexlemp1rp  10670  resqrexlemcalc2  10679  resqrexlemcalc3  10680  resqrexlemgt0  10684  resqrexlemglsq  10686  rpsqrtcl  10705  absrpclap  10725  rpmaxcl  10887  rpmincl  10901  xrminrpcl  10935  xrbdtri  10937  mulcn2  10973  reccn2ap  10974  climge0  10986  divcnv  11158  georeclim  11174  cvgratnnlembern  11184  cvgratnnlemsumlt  11189  cvgratnnlemfm  11190  cvgratnnlemrate  11191  cvgratnn  11192  cvgratz  11193  rpefcl  11242  efltim  11255  ef01bndlem  11314  bdmopn  12493  mulcncflem  12576
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