Theorem elrpii 9808

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 9807	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	mpbir2an 945	1 |- A e. RR+

Syntax hints:   e. wcel 2177   class class class wbr 4054   RRcr 7954   0cc0 7955   < clt 8137   RR+crp 9805

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-rp 9806

This theorem is referenced by:  1rp  9809  2rp  9810  3rp  9811  resqrexlemnm  11414  resqrexlemga  11419  epr  12178  pirp  15346  coseq0negpitopi  15393  pigt3  15401