
theorem Th84: :: Mycielskian0:
for G being SimpleGraph holds G c= Mycielskian G
proof
 let G be SimpleGraph;
 set MG = Mycielskian G; set uG = union G;
 let x be object;
 assume x in G;
  then x in  { {} } \/ singletons uG \/ Edges G by Th27;
  then A1: x in  { {} } \/ singletons uG or x in Edges G by XBOOLE_0:def 3;
  per cases by A1,XBOOLE_0:def 3;
  suppose x in { {} };
     then x = {} by TARSKI:def 1;
    hence x in MG by Th20;
  end;
  suppose x in singletons uG;
     then consider f being Subset of uG such that
  A2: x = f and
  A3: f is 1-element;
     consider a being set such that
  A4: a in uG and
  A5: f = {a} by A3,Th9;
     a in uG \/ [:uG,{uG}:] by A4,XBOOLE_0:def 3;
     then a in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
     then x in the set of all
 {xx} where xx is Element of  uG \/ [:uG,{uG}:] \/ {uG}
              by A2,A5;
    hence x in MG by MYCIELSK:4;
  end;
  suppose A6: x in Edges G;
    Edges G c= MG by MYCIELSK:3;
   hence x in MG by A6;
  end;
end;
