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Theorem elrpii 8586
 Description: Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
Hypotheses
Ref Expression
elrpi.1 𝐴 ∈ ℝ
elrpi.2 0 < 𝐴
Assertion
Ref Expression
elrpii 𝐴 ∈ ℝ+

Proof of Theorem elrpii
StepHypRef Expression
1 elrpi.1 . 2 𝐴 ∈ ℝ
2 elrpi.2 . 2 0 < 𝐴
3 elrp 8585 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
41, 2, 3mpbir2an 849 1 𝐴 ∈ ℝ+
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393   class class class wbr 3764  ℝcr 6888  0cc0 6889   < clt 7060  ℝ+crp 8583 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-rp 8584 This theorem is referenced by:  1rp  8587  2rp  8588  resqrexlemnm  9616  resqrexlemga  9621
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