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Theorem rpgt0 9199
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 9190 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21simprbi 270 1 (𝐴 ∈ ℝ+ → 0 < 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1439   class class class wbr 3851  cr 7403  0cc0 7404   < clt 7576  +crp 9188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-rp 9189
This theorem is referenced by:  rpge0  9200  rpap0  9204  rpgecl  9216  0nrp  9221  rpgt0d  9230  addlelt  9293  rpsqrtcl  10528  climconst  10732
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