Theorem elrpii 9890

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

Step	Hyp	Ref	Expression
1		elrpi.1	. 2 |- A e. RR
2		elrpi.2	. 2 |- 0 < A
3		elrp 9889	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	mpbir2an 950	1 |- A e. RR+

Syntax hints:   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:  1rp  9891  2rp  9892  3rp  9893  resqrexlemnm  11578  resqrexlemga  11583  epr  12342  pirp  15512  coseq0negpitopi  15559  pigt3  15567