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Theorem elrpii 9456
Description: Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
Hypotheses
Ref Expression
elrpi.1  |-  A  e.  RR
elrpi.2  |-  0  <  A
Assertion
Ref Expression
elrpii  |-  A  e.  RR+

Proof of Theorem elrpii
StepHypRef Expression
1 elrpi.1 . 2  |-  A  e.  RR
2 elrpi.2 . 2  |-  0  <  A
3 elrp 9455 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
41, 2, 3mpbir2an 926 1  |-  A  e.  RR+
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   class class class wbr 3929   RRcr 7631   0cc0 7632    < clt 7812   RR+crp 9453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-rp 9454
This theorem is referenced by:  1rp  9457  2rp  9458  3rp  9459  resqrexlemnm  10802  resqrexlemga  10807  epr  11499  pirp  12892  coseq0negpitopi  12939  pigt3  12947
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