reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem Th21:
  not ex X st for x holds x in X iff x is Ordinal
proof
  given X such that
A1: for x holds x in X iff x is Ordinal;
A2: X is epsilon-transitive
  proof
    let x;
    assume x in X;
    then
A3: x is Ordinal by A1;
    thus thesis
    proof
      let a be object;
      assume a in x;
      then a is Ordinal by A3,Th9;
      hence thesis by A1;
    end;
  end;
  X is epsilon-connected
  proof
    let x,y;
    assume x in X & y in X;
    then x is Ordinal & y is Ordinal by A1;
    hence thesis by Th10;
  end;
  then X in X by A1,A2;
  hence contradiction;
end;
