consider x being non empty set such that
A1: x in X by SETFAM_1:def 10;
consider y being object such that
A2: y in x by XBOOLE_0:def 1;
{} c= card n ;
then {} in card n by CARD_1:3;
then 1 c= card n by CARD_2:68;
then A3: card {y} c= card n by CARD_1:30;
{y} c= x by A2, ZFMISC_1:31;
then {y} in X by A1, CLASSES1:def 1;
then {y} in the_subsets_with_limited_card (n,X,D) by A3, Def2;
hence not the_subsets_with_limited_card (n,X,D) is empty-membered ; :: thesis: verum