reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem Th7:
  for PA being a_partition of Y holds Y is_a_dependent_set_of PA
proof
  let PA be a_partition of Y;
A1: {Y} is Subset-Family of Y by ZFMISC_1:68;
A2: union {Y} = Y by ZFMISC_1:25;
   for
 A st A in {Y} holds A<>{} & for B st B in {Y} holds A = B or A misses B
  proof
    let A;
    assume
A3: A in {Y};
    then A4: A=Y by TARSKI:def 1;
    thus A<>{} by A3,TARSKI:def 1;
    let B;
    assume B in {Y};
    hence thesis by A4,TARSKI:def 1;
  end;
then A5: {Y} is a_partition of Y by A1,A2,EQREL_1:def 4;
 for a being set st a in PA ex b being set st b in {Y} & a c= b
  proof
    let a be set;
    assume
A6: a in PA; then
A7: a<>{} by EQREL_1:def 4;
    set x = the Element of a;
    x in Y by A6,A7,TARSKI:def 3;
    then consider b being set such that
    x in b and
A8: b in {Y} by A2,TARSKI:def 4;
 b = Y by A8,TARSKI:def 1;
    hence thesis by A6,A8;
  end;
then A9: {Y} '>' PA by SETFAM_1:def 2;
 Y in {Y} by TARSKI:def 1;
  hence thesis by A5,A9,Th6;
end;
