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Theorem elrpd 9161
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 9126 . 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 1438   class class class wbr 3843   RRcr 7339   0cc0 7340    < clt 7512   RR+crp 9124
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 3450  df-pr 3451  df-op 3453  df-br 3844  df-rp 9125
This theorem is referenced by:  zltaddlt1le  9413  modqval  9719  ltexp2a  9995  leexp2a  9996  expnlbnd2  10067  resqrexlem1arp  10426  resqrexlemp1rp  10427  resqrexlemcalc2  10436  resqrexlemcalc3  10437  resqrexlemgt0  10441  resqrexlemglsq  10443  rpsqrtcl  10462  absrpclap  10482  mulcn2  10688  climge0  10700  divcnv  10878  georeclim  10894  cvgratnnlembern  10904  cvgratnnlemsumlt  10909  cvgratnnlemfm  10910  cvgratnnlemrate  10911  cvgratnn  10912  cvgratz  10913  rpefcl  10962  efltim  10975  ef01bndlem  11034
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