
theorem
  for x,y being object holds {x,y}\{the Element of {x,y}} = {} iff x = y
proof
  let x,y be object;
  set z = the Element of {x,y};
  hereby
    assume {x,y}\{z} = {};
    then {x,y} c= {z} by XBOOLE_1:37;
    then x = z & y = z by ZFMISC_1:20;
    hence x = y;
  end;
  assume x = y;
  then {x,y} = {x} by ENUMSET1:29;
  then {z} = {x,y} by TARSKI:def 1;
  hence thesis by XBOOLE_1:37;
end;
