A1: union PA = Y by EQREL_1:def 4;
A2: for A being set st A in PA holds
( A <> {} & ( for B being set holds
( not B in PA or A = B or A misses B ) ) ) by EQREL_1:def 4;
defpred S₁[ set ] means $1 is_min_depend PA,PB;
consider X being set such that
A3: for d being set holds
( d in X iff ( d in bool Y & S₁[d] ) ) from XBOOLE_0:sch 1();
A4: for d being set holds
( d in X iff d is_min_depend PA,PB )

proof

let d be set ; :: thesis: ( d in X iff d is_min_depend PA,PB )
for d being set st d is_min_depend PA,PB holds
d in bool Y

proof

let d be set ; :: thesis: ( d is_min_depend PA,PB implies d in bool Y )
assume d is_min_depend PA,PB ; :: thesis: d in bool Y
then d is_a_dependent_set_of PA by Def2;
then ex A being set st
( A c= PA & A <> {} & d = union A ) by Def1;
then d c= union PA by ZFMISC_1:77;
hence d in bool Y by A1; :: thesis: verum

end;

then ( d is_min_depend PA,PB implies ( d is_min_depend PA,PB & d in bool Y ) ) ;
hence ( d in X iff d is_min_depend PA,PB ) by A3; :: thesis: verum

end;

take X ; :: thesis: ( X is Element of bool (bool Y) & X is a_partition of Y & ( for d being set holds
( d in X iff d is_min_depend PA,PB ) ) )
A5: Y c= union X

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Y or y in union X )
assume y in Y ; :: thesis: y in union X
then consider a being Subset of Y such that
A6: y in a and
A7: a is_min_depend PA,PB by Th13;
a in X by A4, A7;
hence y in union X by A6, TARSKI:def 4; :: thesis: verum

end;

union X c= Y

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in union X or y in Y )
assume y in union X ; :: thesis: y in Y
then consider a being set such that
A8: y in a and
A9: a in X by TARSKI:def 4;
a is_min_depend PA,PB by A4, A9;
then a is_a_dependent_set_of PA by Def2;
then ex A being set st
( A c= PA & A <> {} & a = union A ) by Def1;
then a c= Y by A1, ZFMISC_1:77;
hence y in Y by A8; :: thesis: verum

end;

then A10: union X = Y by A5, XBOOLE_0:def 10;
A11: for A being Subset of Y st A in X holds
( A <> {} & ( for B being Subset of Y holds
( not B in X or A = B or A misses B ) ) )

proof

let A be Subset of Y; :: thesis: ( A in X implies ( A <> {} & ( for B being Subset of Y holds
( not B in X or A = B or A misses B ) ) ) )
assume A in X ; :: thesis: ( A <> {} & ( for B being Subset of Y holds
( not B in X or A = B or A misses B ) ) )
then A12: A is_min_depend PA,PB by A4;
then A13: A is_a_dependent_set_of PA by Def2;
A14: A is_a_dependent_set_of PB by A12, Def2;
consider Aa being set such that
A15: Aa c= PA and
A16: Aa <> {} and
A17: A = union Aa by A13, Def1;
consider aa being set such that
A18: aa in Aa by A16, XBOOLE_0:def 1;
A19: aa <> {} by A2, A15, A18;
aa c= union Aa by A18, ZFMISC_1:74;
hence A <> {} by A17, A19; :: thesis: for B being Subset of Y holds
( not B in X or A = B or A misses B )
let B be Subset of Y; :: thesis: ( not B in X or A = B or A misses B )
assume B in X ; :: thesis: ( A = B or A misses B )
then A20: B is_min_depend PA,PB by A4;
then A21: ( B is_a_dependent_set_of PA & B is_a_dependent_set_of PB ) by Def2;

now

assume A meets B ; :: thesis: A = B
then A22: ( A /\ B is_a_dependent_set_of PA & A /\ B is_a_dependent_set_of PB ) by A13, A14, A21, Th11;
A /\ B c= A by XBOOLE_1:17;
then A23: A /\ B = A by A12, A22, Def2;
A24: A c= B

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in A or x in B )
assume x in A ; :: thesis: x in B
hence x in B by A23, XBOOLE_0:def 4; :: thesis: verum

end;

A /\ B c= B by XBOOLE_1:17;
then A25: A /\ B = B by A20, A22, Def2;
B c= A

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in B or x in A )
assume x in B ; :: thesis: x in A
hence x in A by A25, XBOOLE_0:def 4; :: thesis: verum

end;

hence A = B by A24, XBOOLE_0:def 10; :: thesis: verum

end;

hence ( A = B or A misses B ) ; :: thesis: verum

end;

X c= bool Y

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in bool Y )
assume x in X ; :: thesis: x in bool Y
then x is_min_depend PA,PB by A4;
then x is_a_dependent_set_of PA by Def2;
then ex a being set st
( a c= PA & a <> {} & x = union a ) by Def1;
then x c= Y by A1, ZFMISC_1:77;
hence x in bool Y ; :: thesis: verum

end;

then reconsider X = X as Subset-Family of Y ;
X is a_partition of Y by A10, A11, EQREL_1:def 4;
hence ( X is Element of bool (bool Y) & X is a_partition of Y & ( for d being set holds
( d in X iff d is_min_depend PA,PB ) ) ) by A4; :: thesis: verum