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Theorem rpgt0 9697
Description: A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
Assertion
Ref Expression
rpgt0 (𝐴 ∈ ℝ+ → 0 < 𝐴)

Proof of Theorem rpgt0
StepHypRef Expression
1 elrp 9687 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21simprbi 275 1 (𝐴 ∈ ℝ+ → 0 < 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160   class class class wbr 4018  cr 7841  0cc0 7842   < clt 8023  +crp 9685
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-rp 9686
This theorem is referenced by:  rpge0  9698  rpap0  9702  rpgecl  9714  0nrp  9721  rpgt0d  9731  addlelt  9800  rpsqrtcl  11085  rpmaxcl  11267  rpmincl  11281  xrminrpcl  11317  climconst  11333  blcntrps  14392  blcntr  14393  bdmet  14479  bdmopn  14481  reeff1o  14671  coseq00topi  14733  coseq0negpitopi  14734
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