per cases ( Y = {} or Y <> {} ) ;

suppose Y = {} ; :: thesis: not PARTITIONS Y is empty

hence not PARTITIONS Y is empty by Def3, EQREL_1:45; :: thesis: verum

end;

suppose Y <> {} ; :: thesis: not PARTITIONS Y is empty

then reconsider Y = Y as non empty set ;
A1: ( {Y} is Subset-Family of Y & union {Y} = Y ) by ZFMISC_1:25, ZFMISC_1:68;
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 A2: A in {Y} ; :: thesis: ( A <> {} & ( for B being Subset of Y holds
( not B in {Y} or A = B or A misses B ) ) )
then A3: A = Y by TARSKI:def 1;
thus A <> {} by A2, 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 A3, TARSKI:def 1; :: thesis: verum

end;

then {Y} is a_partition of Y by A1, EQREL_1:def 4;
hence not PARTITIONS Y is empty by Def3; :: thesis: verum

end;

end;