Theorem elrpii 9780

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 9779	. 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
4	1, 2, 3	mpbir2an 945	1 |- A e. RR+

Syntax hints:   e. wcel 2176   class class class wbr 4045   RRcr 7926   0cc0 7927   < clt 8109   RR+crp 9777

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-rp 9778

This theorem is referenced by:  1rp  9781  2rp  9782  3rp  9783  resqrexlemnm  11362  resqrexlemga  11367  epr  12126  pirp  15294  coseq0negpitopi  15341  pigt3  15349