Theorem rpregt0 9859

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 9847	. 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 2200   class class class wbr 4082   RRcr 7994   0cc0 7995   < clt 8177   RR+crp 9845

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-rp 9846

This theorem is referenced by:  rpne0  9861  divlt1lt  9916  divle1le  9917  ledivge1le  9918  nnledivrp  9958  expnlbnd  10881  isprm6  12664