reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
 B in [:A,B:] implies ex x being object st x in A & B = [x,{x}]
  proof
   assume B in [:A,B:];
    then consider x,y being object such that
A1:   x in A & y in B & B = [x,y] by Th83;
   take x;
   thus x in A by A1;
   per cases by A1,TARSKI:def 2;
   suppose y = {x};
    hence thesis by A1;
   end;
   suppose
A2:   y = {x,y};
     reconsider y as set by TARSKI:1;
     y in y by TARSKI:def 2,A2;
    hence thesis;
   end;
  end;
