
theorem Th31:
for X being set ex G being SimpleGraph st G is edgeless & Vertices G = X
proof
  let X be set;
  set G = {{}} \/ singletons X;
A1:  G is subset-closed proof
      let x,y be set; assume that
    A2: x in G and
    A3: y c= x;
    per cases by A2,XBOOLE_0:def 3;
    suppose x in {{}};
       then x = {} by TARSKI:def 1;
       then y = {} by A3;
        then y in {{}} by TARSKI:def 1;
        hence y in G by XBOOLE_0:def 3;
    end;
    suppose x in singletons X;
       then consider f being Subset of X such that
     A4: x = f and
     A5: f is 1-element;
         consider v being set such that v in X and
     A6: f = {v} by A5,Th9;
       per cases by A3,A4,A6,ZFMISC_1:33;
       suppose y = {};
         then y in {{}} by TARSKI:def 1;
        hence y in G by XBOOLE_0:def 3;
       end;
       suppose y = {v};
        hence y in G by A2,A6,A4;
       end;
    end;
    end;
A7:  G is 1-at_most_dimensional proof
      let x be set;
      assume A8: x in G;
      per cases by A8,XBOOLE_0:def 3;
      suppose x in {{}};
        then x = {} by TARSKI:def 1;
       hence card x c= 1+1;
    end;
    suppose x in singletons X;
       then consider f being Subset of X such that
     A9: x = f and
     A10: f is 1-element;
         consider v being set such that v in X and
     A11: f = {v} by A10,Th9;
A12:     card x = 1 by A9,A11,CARD_1:30;
       Segm 1 c= Segm(1+1) by NAT_1:39;
      hence card x c= 1+1 by A12;
     end;
    end;
  reconsider G as SimpleGraph by A1,A7;
  take G;
  now
   assume Edges G <> {};
    then consider e being object such that
   A13: e in Edges G by XBOOLE_0:def 1;
   reconsider e as set by TARSKI:1;
   A14: e in G & card e = 2 by A13,Def1;
   per cases by A13,XBOOLE_0:def 3;
   suppose e in {{}};
    hence contradiction by A14,CARD_1:27,TARSKI:def 1;
   end;
   suppose e in singletons X;
       then consider f being Subset of X such that
     A15: e = f and
     A16: f is 1-element;
         consider v being set such that v in X and
     A17: f = {v} by A16,Th9;
     thus contradiction by A14,A15,A17,CARD_1:30;
     end;
  end;
  hence G is edgeless;
  thus Vertices G = X proof
   thus Vertices G c= X proof
     let x be object;
     assume x in Vertices G;
       then consider y being set such that
   A18: x in y and
   A19: y in G by TARSKI:def 4;
   per cases by A19,XBOOLE_0:def 3;
   suppose y in {{}};
    hence thesis by A18,TARSKI:def 1;
   end;
   suppose y in singletons X;
       then consider f being Subset of X such that
     A20: y = f and
         f is 1-element;
     thus x in X by A20,A18;
   end;
   end;
   thus X c= Vertices G proof
     let x be object;
     assume x in X;
       then reconsider f = {x} as Subset of X by ZFMISC_1:31;
       f is 1-element;
     then {x} in singletons X;
     then {x} in G by XBOOLE_0:def 3;
    hence x in Vertices G by Th24;
   end;
  end;
end;
