theorem :: SETFAM_1:5
  EmptyMS I in SF implies meet SF = EmptyMS I
proof
  assume
A1: EmptyMS I in SF;
  now
    let i be object;
    assume
A2: i in I;
    then consider Q be Subset-Family of (M.i) such that
A3: Q = SF.i and
A4: (meet SF).i = Intersect Q by Def1;
    EmptyMS I.i in Q by A1,A2,A3;
    then
A5: {} in Q;
    EmptyMS I.i in SF.i by A1,A2;
    then Intersect Q = meet Q by A3,SETFAM_1:def 9;
    then Intersect Q = {} by A5,SETFAM_1:4;
    hence (meet SF).i = EmptyMS I.i by A4;
  end;
  hence thesis;
end;
