Theorem elrpi 6284

Description: Membership in the set of positive reals.

Hypotheses

Ref	Expression
elrpi.1	|- A e. RR
elrpi.2	|- 0 < A

Assertion

Ref	Expression
elrpi	|- A e. RR+

Proof of Theorem elrpi

Step	Hyp	Ref	Expression
1		elrp 6283	. 2 |- (A e. RR+ <-> (A e. RR /\ 0 < A))
2		elrpi.1	. 2 |- A e. RR
3		elrpi.2	. 2 |- 0 < A
4	1, 2, 3	mpbir2an 732	1 |- A e. RR+

Syntax hints:   e. wcel 960   class class class wbr 2624  RRcr 5245  0cc0 5246  RR+crp 5312   < clt 5498

This theorem is referenced by:  rpexpclt 6583  mulc1cncf 7279  minveclem25 8565  minveclem26 8566  minveclem27 8567  pilem3 8668  log1 8761  loge 8762  dmse1 10594  iintlem1 10603  iintlem2 10604

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-rp 6282

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