reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th9:
  G is edgeless iff G.allSG() = the set of all createGraph(V)
    where V is non empty Subset of the_Vertices_of G
proof
  set S = the set of all createGraph(V)
    where V is non empty Subset of the_Vertices_of G;
  hereby
    assume A1: G is edgeless;
    now
      let x be object;
      hereby
        assume x in G.allSG();
        then reconsider H = x as plain Subgraph of G by Th1;
        reconsider V = [#]the_Vertices_of H
          as non empty Subset of the_Vertices_of G;
        H == createGraph([#]the_Vertices_of H) by A1, Th7;
        then H = createGraph(V) by GLIB_009:44;
        hence x in S;
      end;
      assume x in S;
      then consider V being non empty Subset of the_Vertices_of G such that
        A2: x = createGraph(V);
      thus x in G.allSG() by A2, Th4;
    end;
    hence G.allSG() = S by TARSKI:2;
  end;
  assume G.allSG() = S;
  then G | _GraphSelectors in S by Th3;
  then consider V being non empty Subset of the_Vertices_of G such that
    A3: G | _GraphSelectors = createGraph(V);
  G == G | _GraphSelectors by GLIB_000:128;
  hence G is edgeless by A3, GLIB_008:52;
end;
