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 = {} or {x,y} \ X = {x} or {x,y} \ X = {y} or {x,y} \ X = {x ,y}
proof
  assume that
A1: {x,y} \ X <> {} and
A2: {x,y} \ X <> {x} and
A3: {x,y} \ X <> {y};
A4: not x in X & x<>y or y in X by A3,Lm11;
  x in X or not y in X & x<>y by A2,Lm11;
  hence thesis by A1,A4,Th50,Th63,XBOOLE_1:83;
end;
