thus ( G <> {} implies ex X being a_partition of Y st
for x being set holds
( x in X iff x is_upper_min_depend_of G ) ) :: thesis: ( not G <> {} implies ex b₁ being a_partition of Y st b₁ = %I Y )

proof

defpred S₁[ set ] means $1 is_upper_min_depend_of G;
consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool Y & S₁[x] ) ) from XFAMILY:sch 1();
A2: union X c= Y

proof

let y be object ; :: 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
A3: y in a and
A4: a in X by TARSKI:def 4;
a in bool Y by A1, A4;
hence y in Y by A3; :: thesis: verum

end;

assume A5: G <> {} ; :: thesis: ex X being a_partition of Y st
for x being set holds
( x in X iff x is_upper_min_depend_of G )
then consider g being object such that
A6: g in G by XBOOLE_0:def 1;
reconsider g = g as a_partition of Y by A6, PARTIT1:def 3;
A7: union g = Y by EQREL_1:def 4;
A8: for x being set holds
( x in X iff x is_upper_min_depend_of G )

proof

let x be set ; :: thesis: ( x in X iff x is_upper_min_depend_of G )
for x being set st x is_upper_min_depend_of G holds
x in bool Y

proof

let x be set ; :: thesis: ( x is_upper_min_depend_of G implies x in bool Y )
assume x is_upper_min_depend_of G ; :: thesis: x in bool Y
then ex A being set st
( A c= g & A <> {} & x = union A ) by A6, PARTIT1:def 1;
then x c= union g by ZFMISC_1:77;
hence x in bool Y by A7; :: thesis: verum

end;

hence ( x in X iff x is_upper_min_depend_of G ) by A1; :: thesis: verum

end;

A9: Y c= union X

proof

let y be object ; :: 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 reconsider y = y as Element of Y ;
consider a being Subset of Y such that
A10: y in a and
A11: a is_upper_min_depend_of G by A5, Th2;
a in X by A8, A11;
hence y in union X by A10, TARSKI:def 4; :: thesis: verum

end;

A12: for A being set st A in g holds
( A <> {} & ( for B being set holds
( not B in g or A = B or A misses B ) ) ) by EQREL_1:def 4;
A13: 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 A14: A is_upper_min_depend_of G by A8;
then consider Aa being set such that
A15: Aa c= g and
A16: Aa <> {} and
A17: A = union Aa by A6, PARTIT1:def 1;
consider aa being object such that
A18: aa in Aa by A16, XBOOLE_0:def 1;
reconsider aa = aa as set by TARSKI:1;
A19: aa c= union Aa by A18, ZFMISC_1:74;
aa <> {} by A12, A15, A18;
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_upper_min_depend_of G by A8;

now :: thesis: ( A meets B implies A = B )

assume A21: A meets B ; :: thesis: A = B
A22: for d being a_partition of Y st d in G holds
A /\ B is_a_dependent_set_of d

proof

let d be a_partition of Y; :: thesis: ( d in G implies A /\ B is_a_dependent_set_of d )
assume d in G ; :: thesis: A /\ B is_a_dependent_set_of d
then ( A is_a_dependent_set_of d & B is_a_dependent_set_of d ) by A14, A20;
hence A /\ B is_a_dependent_set_of d by A21, PARTIT1:9; :: thesis: verum

end;

A23: A /\ B = B by A20, A22, XBOOLE_1:17;
A24: B c= A by A23, XBOOLE_0:def 4;
A25: A /\ B = A by A14, A22, XBOOLE_1:17;
A c= B by A25, XBOOLE_0:def 4;
hence A = B by A24, XBOOLE_0:def 10; :: thesis: verum

end;

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

end;

take X ; :: thesis: ( X is Element of bool (bool Y) & X is a_partition of Y & ( for x being set holds
( x in X iff x is_upper_min_depend_of G ) ) )
X c= bool Y by A1;
then reconsider X = X as Subset-Family of Y ;
X is a_partition of Y by A9, A13, A2, EQREL_1:def 4, XBOOLE_0:def 10;
hence ( X is Element of bool (bool Y) & X is a_partition of Y & ( for x being set holds
( x in X iff x is_upper_min_depend_of G ) ) ) by A8; :: thesis: verum

end;

thus ( not G <> {} implies ex b₁ being a_partition of Y st b₁ = %I Y ) ; :: thesis: verum