
theorem Th85:
for G being SimpleGraph, v being set
 holds v in Vertices Mycielskian G
   iff v in union G or (ex x being set st x in union G & v = [x,union G])
       or v = union G
proof
  let G be SimpleGraph, v be set;
  set uG = union G; set MG = Mycielskian G; set uMG = union MG;
  hereby
  assume v in Vertices MG;
  then consider g being set such that
A1: v in g and
A2: g in MG by TARSKI:def 4;
  defpred Thesis[] means
          v in union G or (ex x being set st x in union G & v = [x,union G])
       or v = union G;
  per cases by A2,MYCIELSK:4;
  suppose g in { {} };
   hence Thesis[] by A1,TARSKI:def 1;
  end;
  suppose g in the set of all
 {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG};
     then consider h being Element of uG \/ [:uG,{uG}:] \/ {uG} such that
  A3: g = {h};
  A4: h in uG \/ [:uG,{uG}:] or h in {uG} by XBOOLE_0:def 3;
  A5: v = h by A3,A1,TARSKI:def 1;
     per cases by A4,XBOOLE_0:def 3;
     suppose h in uG;
      hence Thesis[] by A3,A1,TARSKI:def 1;
     end;
     suppose h in [:uG,{uG}:];
       then consider h1, h2 being object such that
    A6: h1 in uG and
    A7: h2 in {uG} and
    A8: h = [h1,h2] by ZFMISC_1:def 2;
        h2 = uG by A7,TARSKI:def 1;
      hence Thesis[] by A5,A8,A6;
     end;
     suppose h in {uG};
      hence Thesis[] by A5,TARSKI:def 1;
     end;
  end;
  suppose g in Edges G;
     then consider g1, g2 being set such that g1 <> g2 and
  A9: g1 in Vertices G and
  A10: g2 in Vertices G and
  A11: g = {g1, g2} by Th11;
   thus Thesis[] by A9,A10,A1,A11,TARSKI:def 2;
  end;
  suppose g in { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G };
    then consider g1, g2 being Element of uG such that
  A12: g = {g1,[g2,uG]} and
  A13: {g1,g2} in Edges G;
  A14: g1 in uG & g2 in uG by A13,Th13;
      v = g1 or v = [g2,uG] by A12,A1,TARSKI:def 2;
   hence Thesis[] by A14;
  end;
  suppose g in { {uG,[x,uG]} where x is Element of uG : x in Vertices G };
     then consider x being Element of uG such that
  A15: g = {uG,[x,uG]} and
  A16: x in uG;
      v = uG or v = [x,uG] by A1,A15,TARSKI:def 2;
   hence Thesis[] by A16;
  end;
 end;
 assume A17: v in union G or (ex x being set st x in union G & v = [x,union G])
        or v = union G;
A18: for a being set st a in uG \/ [:uG,{uG}:] \/ {uG} holds a in uMG
   proof
     let a be set;
     assume a in uG \/ [:uG,{uG}:] \/ {uG};
     then A19: {a} in the set of all
 {x} where x is Element of uG \/ [:uG,{uG}:] \/ {uG};
   A20: the set of all {x} where x is Element of uG \/ [:uG,{uG}:] \/ {uG}
            c= MG by MYCIELSK:3;
      a in {a} by TARSKI:def 1;
     hence a in uMG by A20,A19,TARSKI:def 4;
   end;
 per cases by A17;
 suppose v in union G;
    then v in uG \/ [:uG,{uG}:] by XBOOLE_0:def 3;
    then v in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
   hence thesis by A18;
  end;
 suppose (ex x being set st x in union G & v = [x,union G]);
  then consider x being set such that
 A21: x in union G and
 A22: v = [x,union G];
  union G in {union G} by TARSKI:def 1;
  then v in [:uG,{uG}:] by A21,A22,ZFMISC_1:def 2;
    then v in uG \/ [:uG,{uG}:] by XBOOLE_0:def 3;
    then v in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
   hence thesis by A18;
  end;
 suppose v = union G; then
    v in {uG} by TARSKI:def 1;
    then v in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
   hence thesis by A18;
 end;
end;
