let Y be non empty set ; :: thesis: for PA being a_partition of Y holds Y is_a_dependent_set_of PA
let PA be a_partition of Y; :: thesis: Y is_a_dependent_set_of PA
A1: {Y} is Subset-Family of Y by ZFMISC_1:80;
A2: union {Y} = Y by ZFMISC_1:31;
for A being Subset of Y st A in {Y} holds
( A <> {} & ( for B being Subset of Y holds
( not B in {Y} or A = B or A misses B ) ) )

proof

let A be Subset of Y; :: thesis: ( A in {Y} implies ( A <> {} & ( for B being Subset of Y holds
( not B in {Y} or A = B or A misses B ) ) ) )
assume A3: A in {Y} ; :: thesis: ( A <> {} & ( for B being Subset of Y holds
( not B in {Y} or A = B or A misses B ) ) )
then A4: A = Y by TARSKI:def 1;
thus A <> {} by A3, TARSKI:def 1; :: thesis: for B being Subset of Y holds
( not B in {Y} or A = B or A misses B )
let B be Subset of Y; :: thesis: ( not B in {Y} or A = B or A misses B )
assume B in {Y} ; :: thesis: ( A = B or A misses B )
hence ( A = B or A misses B ) by A4, TARSKI:def 1; :: thesis: verum

end;

then A5: {Y} is a_partition of Y by A1, A2, EQREL_1:def 6;
A6: {Y} '>' PA

proof

for a being set st a in PA holds
ex b being set st
( b in {Y} & a c= b )

proof

let a be set ; :: thesis: ( a in PA implies ex b being set st
( b in {Y} & a c= b ) )
assume A7: a in PA ; :: thesis: ex b being set st
( b in {Y} & a c= b )
then A8: ( a c= Y & a <> {} ) by EQREL_1:def 6;
consider x being Element of a;
x in Y by A8, TARSKI:def 3;
then consider b being set such that
A9: ( x in b & b in {Y} ) by A2, TARSKI:def 4;
b = Y by A9, TARSKI:def 1;
hence ex b being set st
( b in {Y} & a c= b ) by A7, A9; :: thesis: verum

end;

hence {Y} '>' PA by SETFAM_1:def 2; :: thesis: verum

end;

Y in {Y} by TARSKI:def 1;
hence Y is_a_dependent_set_of PA by A5, A6, Th8; :: thesis: verum