
theorem
  for a,b,x,y being object st a <> b & x = the Element of {a,b} &
    y = the Element of ({a,b}\{the Element of {a,b}})
  holds x = a & y = b or x = b & y = a
proof
  let a,b,x,y be object;
  assume A1: a <> b & x = the Element of {a,b} &
    y = the Element of ({a,b}\{the Element of {a,b}});
  then per cases by TARSKI:def 2;
  suppose A2: x = a;
    {a,b}\{a} = {b} by A1, ZFMISC_1:17;
    hence thesis by A1, A2, TARSKI:def 1;
  end;
  suppose A3: x = b;
    {a,b}\{b} = {a} by A1, ZFMISC_1:17;
    hence thesis by A1, A3, TARSKI:def 1;
  end;
end;
