Theorem rpregt0 9901

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 9889	. 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 2202   class class class wbr 4088   RRcr 8030   0cc0 8031   < clt 8213   RR+crp 9887

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-rp 9888

This theorem is referenced by:  rpne0  9903  divlt1lt  9958  divle1le  9959  ledivge1le  9960  nnledivrp  10000  expnlbnd  10925  isprm6  12718