consider x being set such that
A1:
x in D
by XBOOLE_0:def 1;
{x} c= D
by A1, ZFMISC_1:37;
then reconsider XX = bool {x} as Subset-Family of D by ZFMISC_1:79;
take
XX
; ( XX is finite & not XX is empty-membered & XX is subset-closed & XX is finite-membered )
{x} in bool {x}
by ZFMISC_1:def 1;
hence
( XX is finite & not XX is empty-membered & XX is subset-closed & XX is finite-membered )
by SETFAM_1:def 11; verum