reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th11:
  (for Y st Y in F holds Y is Dynkin_System of Omega) implies meet
  F is Dynkin_System of Omega
proof
  assume
A1: for Y st Y in F holds Y is Dynkin_System of Omega;
  now
    let Y;
    assume Y in F;
    then Y is Dynkin_System of Omega by A1;
    hence {} in Y by Def5;
  end;
  then
A2: {} in meet F by SETFAM_1:def 1;
A3: now
    let f;
    assume that
A4: rng f c= meet F and
A5: f is disjoint_valued;
    now
      let Y such that
A6:   Y in F;
      meet F c= Y by A6,SETFAM_1:3;
      then
A7:   rng f c= Y by A4;
      Y is Dynkin_System of Omega by A1,A6;
      hence Union f in Y by A5,A7,Def5;
    end;
    hence Union f in meet F by SETFAM_1:def 1;
  end;
A8: now
    let X;
    assume
A9: X in meet F;
    for Y st Y in F holds X` in Y
    proof
      let Y;
      assume Y in F;
      then Y is Dynkin_System of Omega & meet F c= Y by A1,SETFAM_1:3;
      hence thesis by A9,Def5;
    end;
    hence X` in meet F by SETFAM_1:def 1;
  end;
  set Y = the Element of F;
A10: meet F c= Y by SETFAM_1:3;
  Y is Dynkin_System of Omega by A1;
  then meet F is Subset-Family of Omega by A10,XBOOLE_1:1;
  hence thesis by A3,A8,A2,Def5;
end;
