theorem
  X c= [:X,Y:] implies X = {}
proof
  assume
A1: X c= [:X,Y:];
  assume X <> {};
  then consider z such that
A2: z in X by XBOOLE_0:7;
  consider M such that
A3: M in X \/ union X and
A4: X \/ union X misses M by A2,XREGULAR:1;
  now
    assume
A5: M in X;
    then consider x,y such that
A6: M = [x,y] by Lm19,A1;
A7: {x} in M by A6,TARSKI:def 2;
    M c= union X by A5,Lm14;
    then {x} in union X by A7;
    then {x} in X \/ union X by XBOOLE_0:def 3;
    hence contradiction by A4,A7,XBOOLE_0:3;
  end;
  then M in union X by A3,XBOOLE_0:def 3;
  then consider Z such that
A8: M in Z and
A9: Z in X by TARSKI:def 4;
  Z in [:X,Y:] by A1,A9;
  then consider x,y such that
A10: x in X and y in Y and
A11: Z = [x,y] by Def2;
  M={x} or M={x,y} by A8,A11,TARSKI:def 2;
  then
A12: x in M by TARSKI:def 1,def 2;
  x in X \/ union X by A10,XBOOLE_0:def 3;
  hence contradiction by A4,A12,XBOOLE_0:3;
end;
