Theorem rpregt0 9556

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 9544	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
2	1	biimpi 119	1 |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2128   class class class wbr 3965   RRcr 7714   0cc0 7715   < clt 7895   RR+crp 9542

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-rp 9543

This theorem is referenced by:  rpne0  9558  divlt1lt  9613  divle1le  9614  ledivge1le  9615  nnledivrp  9655  expnlbnd  10524  isprm6  12001