ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rpgt0 GIF version

Theorem rpgt0 8826
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 8817 . 2 (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
21simprbi 269 1 (𝐴 ∈ ℝ+ → 0 < 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434   class class class wbr 3793  cr 7042  0cc0 7043   < clt 7215  +crp 8815
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 3412  df-pr 3413  df-op 3415  df-br 3794  df-rp 8816
This theorem is referenced by:  rpge0  8827  rpap0  8831  rpgecl  8843  0nrp  8848  rpgt0d  8857  addlelt  8920  rpsqrtcl  10065  climconst  10267
  Copyright terms: Public domain W3C validator