
theorem
  for G being loopless _Graph holds G is edgeless iff
    for v,w being Vertex of G st v <> w holds not v,w are_adjacent
proof
  let G be loopless _Graph;
  thus G is edgeless implies
    for v,w being Vertex of G st v <> w holds not v,w are_adjacent by Th65;
  assume A1: for v,w being Vertex of G st v <> w holds not v,w are_adjacent;
  assume G is non edgeless;
  then consider e being object such that
    A2: e in the_Edges_of G by XBOOLE_0:def 1;
  set v = (the_Source_of G).e, w = (the_Target_of G).e;
  reconsider v,w as Vertex of G by A2, FUNCT_2:5;
  A3: e Joins v,w,G by A2, GLIB_000:def 13;
  then v <> w by GLIB_000:18;
  hence contradiction by A1, A3, CHORD:def 3;
end;
