
theorem :: MGsingle:
for G being SimpleGraph, x being set st Vertices G = {x}
holds Mycielskian G = {{},{x},{[x,union G]}, {union G}, {[x,union G],union G}}
proof
 let G be SimpleGraph, a be set such that A1: Vertices G = {a};
A2: card Vertices G = 1 by A1,CARD_1:30;
A3: a in Vertices G by A1,TARSKI:def 1;
   set uG = union G; set MG = Mycielskian G;
   set A = the set of all
{x} where x is Element of (union G)\/[:uG,{uG}:] \/ {uG};
   set B = {{x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G };
   set C = { {uG,[x,uG]} where x is Element of uG : x in Vertices G };
   consider aa being object such that
A4: uG = {aa} by A2,CARD_2:42;
A5:  a = aa by A4,A3,TARSKI:def 1;
A6: [:uG,{uG}:] = {[a,uG]} by A4,A5,ZFMISC_1:29;
A7: G is edgeless by A2,Th19;
 thus Mycielskian G c= {{},{a},{[a,uG]}, {uG}, {[a,uG],uG}} proof
   let e be object;
   assume A8: e in MG;
   per cases by A8,MYCIELSK:4;
   suppose e in {{}};
     then e = {} by TARSKI:def 1;
    hence e in {{},{a},{[a,uG]}, {uG}, {[a,uG],uG}} by ENUMSET1:def 3;
   end;
   suppose e in A;
     then consider x being Element of uG \/ [:uG,{uG}:] \/ {uG} such that
   A9: e = {x};
       x in uG \/ [:uG,{uG}:] or x in {uG} by XBOOLE_0:def 3;
       then x in uG or x in [:uG,{uG}:] or x in {uG} by XBOOLE_0:def 3;
       then x = a or x = [a,uG] or x = uG by A4,A5,A6,TARSKI:def 1;
    hence thesis by A9,ENUMSET1:def 3;
   end;
   suppose e in Edges G;
    hence thesis by A7;
   end;
   suppose e in B;
      then consider x, y being Element of uG such that e = {x,[y,uG]} and
   A10: {x,y} in Edges G;
    thus thesis by A10,A7;
   end;
   suppose e in C;
      then consider x being Element of uG such that
   A11: e = {uG,[x,uG]} and x in Vertices G;
       x = a by A4,A5,TARSKI:def 1;
    hence thesis by A11,ENUMSET1:def 3;
   end;
 end;
 thus {{},{a},{[a,union G]}, {union G}, {[a,union G],union G}} c= Mycielskian G
 proof
   let e be object;
   assume A12: e in {{},{a},{[a,union G]}, {union G}, {[a,union G],union G}};
   per cases by A12,ENUMSET1:def 3;
   suppose e = {};
    hence e in MG by Th20;
   end;
   suppose A13: e = {a};
     a in uG \/ [:uG,{uG}:] by A3,XBOOLE_0:def 3;
     then a in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
     then e in A by A13;
    hence e in MG by MYCIELSK:4;
   end;
   suppose A14: e = {[a,union G]};
       [a,union G] in [:uG,{uG}:] by A6,TARSKI:def 1;
       then [a,union G] in uG \/ [:uG,{uG}:] by XBOOLE_0:def 3;
       then [a,union G] in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
       then e in A by A14;
    hence e in MG by MYCIELSK:4;
   end;
   suppose A15: e = {uG};
      uG in {uG} by TARSKI:def 1;
      then uG in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
      then e in A by A15;
    hence e in MG by MYCIELSK:4;
   end;
   suppose e = {[a,uG],uG};
      then e in C by A3;
    hence e in MG by MYCIELSK:4;
   end;
 end;
end;
