Theorem elrpii 9658

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 9657	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	mpbir2an 942	1 |- A e. RR+

Syntax hints:   e. wcel 2148   class class class wbr 4005   RRcr 7812   0cc0 7813   < clt 7994   RR+crp 9655

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-rp 9656

This theorem is referenced by:  1rp  9659  2rp  9660  3rp  9661  resqrexlemnm  11029  resqrexlemga  11034  epr  11791  pirp  14295  coseq0negpitopi  14342  pigt3  14350