Theorem rpregt0 10000

Description: A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Assertion

Ref	Expression
rpregt0	|- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )

Proof of Theorem rpregt0

Step	Hyp	Ref	Expression
1		elrp 9988	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
2	1	biimpi 120	1 |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   class class class wbr 4109   RRcr 8126   0cc0 8127   < clt 8308   RR+crp 9986

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-rp 9987

This theorem is referenced by:  rpne0  10002  divlt1lt  10057  divle1le  10058  ledivge1le  10059  nnledivrp  10099  expnlbnd  11026  isprm6  12844