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
  bool {} = { {} }
proof
  now
    let x;
    reconsider xx = x as set by TARSKI:1;
    xx c= {} iff x = {} by XBOOLE_1:3;
    hence x in bool {} iff x in { {} } by Def1,TARSKI:def 1;
  end;
  hence thesis by TARSKI:2;
end;
