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_1:68;
A2: union {Y} = Y by ZFMISC_1:25;
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 4;
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 A6: a in PA ; :: thesis: ex b being set st
( b in {Y} & a c= b )
then A7: a <> {} by EQREL_1:def 4;
set x = the Element of a;
the Element of a in Y by A6, A7, TARSKI:def 3;
then consider b being set such that
the Element of a in b and
A8: b in {Y} by A2, TARSKI:def 4;
b = Y by A8, TARSKI:def 1;
hence ex b being set st
( b in {Y} & a c= b ) by A6, A8; :: thesis: verum

end;

then A9: {Y} '>' PA by SETFAM_1:def 2;
Y in {Y} by TARSKI:def 1;
hence Y is_a_dependent_set_of PA by A5, A9, Th6; :: thesis: verum