reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for G being Subset of PARTITIONS(Y),PA being a_partition of Y st PA in
  G holds PA '<' ('\/' G)
proof
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  assume
A1: PA in G;
  for a being set st a in PA ex b being set st b in ('\/' G) & a c= b
  proof
    let a be set;
    set x = the Element of a;
A2: union ('\/' G) = Y by EQREL_1:def 4;
    assume
A3: a in PA;
    then
A4: a<>{} by EQREL_1:def 4;
    then x in Y by A3,TARSKI:def 3;
    then consider b being set such that
A5: x in b and
A6: b in ('\/' G) by A2,TARSKI:def 4;
    b is_upper_min_depend_of G by A1,A6,Def3;
    then consider B being set such that
A7: B c= PA and
    B<>{} and
A8: b = union B by A1,PARTIT1:def 1;
    a in B
    proof
      consider u being set such that
A9:   x in u and
A10:  u in B by A5,A8,TARSKI:def 4;
A11:  a /\ u <> {} by A4,A9,XBOOLE_0:def 4;
      u in PA by A7,A10;
      hence thesis by A3,A10,A11,EQREL_1:def 4,XBOOLE_0:def 7;
    end;
    hence thesis by A6,A8,ZFMISC_1:74;
  end;
  hence thesis by SETFAM_1:def 2;
end;
