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_1:31;
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 that
A1: X in S and
A2: Y in S ; :: thesis: ( X is_finer_than Y or Y is_finer_than X )
X = SmallestPartition A by A1, TARSKI:def 1;
hence ( X is_finer_than Y or Y is_finer_than X ) by A2, TARSKI:def 1; :: thesis: verum
end;
hence {(SmallestPartition A)} is Classification of A ; :: thesis: verum