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;

theorem
  {x} \/ {y} = {x} implies x = y
proof
  assume {x} \/ {y} = {x};
  then y in {x} or x in {y} by Lm4;
  hence thesis by TARSKI:def 1;
end;
