
theorem Th16:
  for X being set, A being empty Subset-Family of X holds UniCl A = {{}}
proof
  let X be set, A be empty Subset-Family of X;
  hereby
    let x be object;
    assume x in UniCl A;
    then consider B being Subset-Family of X such that
A1: B c= A and
A2: x = union B by CANTOR_1:def 1;
    B = {} by A1;
    hence x in {{}} by A2,TARSKI:def 1,ZFMISC_1:2;
  end;
  let x be object;
  assume x in {{}};
  then
A3: x = {} by TARSKI:def 1;
  union {}bool X = {} by ZFMISC_1:2;
  hence thesis by A3,CANTOR_1:def 1;
end;
