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Theorem rpgt0d 8846
Description: A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rpred.1 (𝜑𝐴 ∈ ℝ+)
Assertion
Ref Expression
rpgt0d (𝜑 → 0 < 𝐴)

Proof of Theorem rpgt0d
StepHypRef Expression
1 rpred.1 . 2 (𝜑𝐴 ∈ ℝ+)
2 rpgt0 8815 . 2 (𝐴 ∈ ℝ+ → 0 < 𝐴)
31, 2syl 14 1 (𝜑 → 0 < 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434   class class class wbr 3787  0cc0 7032   < clt 7204  +crp 8804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-rp 8805
This theorem is referenced by:  rpregt0d  8850  ltmulgt11d  8879  ltmulgt12d  8880  gt0divd  8881  ge0divd  8882  lediv12ad  8903  expgt0  9595  nnesq  9678  bccl2  9781  resqrexlemp1rp  10019  resqrexlemover  10023  resqrexlemnm  10031  resqrexlemgt0  10033  resqrexlemglsq  10035  sqrtgt0d  10172  prmind2  10635  sqrt2irrlem  10673
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