let A be non empty set ; :: thesis: {(SmallestPartition A)} is Classification of A
SmallestPartition A in PARTITIONS A by PARTIT1:def 3;
then reconsider S = {(SmallestPartition A)} as Subset of (PARTITIONS A) by ZFMISC_1:37;
S is Classification of A

proof

let X, Y be a_partition of A; :: according to TAXONOM1:def 1 :: thesis: ( X in S & Y in S & not X is_finer_than Y implies Y is_finer_than X )
assume ( X in S & Y in S ) ; :: thesis: ( X is_finer_than Y or Y is_finer_than X )
then ( X = SmallestPartition A & Y = SmallestPartition A ) by TARSKI:def 1;
hence ( X is_finer_than Y or Y is_finer_than X ) ; :: thesis: verum

end;

hence {(SmallestPartition A)} is Classification of A ; :: thesis: verum