Theorem elrpi 6283

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 6282	. 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 730	1 |- A e. RR+

Syntax hints:   e. wcel 958   class class class wbr 2619  RRcr 5233  0cc0 5234  RR+crp 5300   < clt 5486

This theorem is referenced by:  rpexpclt 6582  mulc1cncf 7279  minveclem25 8569  minveclem26 8570  minveclem27 8571  pilem3 8673  log1 8766  loge 8767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-rp 6281

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