Theorem elrpii 9881

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 9880	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	mpbir2an 948	1 |- A e. RR+

Syntax hints:   e. wcel 2200   class class class wbr 4086   RRcr 8021   0cc0 8022   < clt 8204   RR+crp 9878

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-rp 9879

This theorem is referenced by:  1rp  9882  2rp  9883  3rp  9884  resqrexlemnm  11569  resqrexlemga  11574  epr  12333  pirp  15503  coseq0negpitopi  15550  pigt3  15558