
theorem Th1:
  for A,x,y being set st A c= {x,y} & x in A & not y in A holds A = {x}
proof
  let A,x,y be set such that
A1: A c= {x,y} and
A2: x in A and
A3: not y in A;
  per cases by A1,ZFMISC_1:36;
  suppose
    A = {};
    hence thesis by A2;
  end;
  suppose
    A = {x};
    hence thesis;
  end;
  suppose
    A = {y};
    hence thesis by A3,TARSKI:def 1;
  end;
  suppose
    A = {x,y};
    hence thesis by A3,TARSKI:def 2;
  end;
end;
