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Theorem rpregt0 9580
Description: A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
rpregt0 (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Proof of Theorem rpregt0
StepHypRef Expression
1 elrp 9568 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21biimpi 119 1 (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2128   class class class wbr 3967  cr 7733  0cc0 7734   < clt 7914  +crp 9566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-rp 9567
This theorem is referenced by:  rpne0  9582  divlt1lt  9637  divle1le  9638  ledivge1le  9639  nnledivrp  9679  expnlbnd  10551  isprm6  12037
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