
theorem Th27:
for G being SimpleGraph holds G = { {} } \/ singletons Vertices G \/ Edges G
proof let G be SimpleGraph;
 thus G c= { {} } \/ singletons Vertices G \/ Edges G proof
  let x be object;
  assume A1: x in G;
    reconsider v = x as finite set by A1;
   card v <= 1+1 by A1,Th21;
   then card v = 0 or ... or card v = 2;
   then per cases;
   suppose card v = 0;
      then v = {};
      then v in {{}} by TARSKI:def 1;
      then v in {{}} \/ singletons Vertices G by XBOOLE_0:def 3;
     hence thesis by XBOOLE_0:def 3;
   end;
   suppose card v = 1;
      then consider a being object such that
   A2: v = {a} by CARD_2:42;
   A3: a in v by A2,TARSKI:def 1;
   A4: a in union G by A3,A1,TARSKI:def 4;
       reconsider v as Subset of Vertices G by A4,A2,ZFMISC_1:31;
       v in singletons Vertices G by A2;
       then v in {{}} \/ singletons Vertices G by XBOOLE_0:def 3;
       hence thesis by XBOOLE_0:def 3;
   end;
   suppose card v = 2;
     then v in Edges G by A1,Def1;
     hence thesis by XBOOLE_0:def 3;
   end;
 end;
 thus { {} } \/ singletons Vertices G \/ Edges G c= G proof
   let x be object;
   assume x in { {} } \/ singletons Vertices G \/ Edges G;
   then A5: x in { {} } \/ singletons Vertices G or x in Edges G
     by XBOOLE_0:def 3;
   per cases by A5,XBOOLE_0:def 3;
   suppose A6: x in { {} };
      consider z being object such that
   A7: z in G by XBOOLE_0:def 1;
    reconsider z as set by TARSKI:1;
   A8: {} c= z;
       x = {} by A6,TARSKI:def 1;
     hence x in G by A8,A7,CLASSES1:def 1;
   end;
   suppose x in singletons Vertices G;
      then consider f being Subset of Vertices G such that
   A9: x = f and
   A10: f is 1-element;
      consider v being set such that
   A11: v in Vertices G and
   A12: f = {v} by A10,Th9;
     thus x in G by A9,A11,A12,Th24;
   end;
   suppose x in Edges G;
     hence x in G;
   end;
 end;
end;
