let A be non empty trivial set ; for B being set st not A /\ B is empty holds
A c= B
let B be set ; ( not A /\ B is empty implies A c= B )
A1:
A /\ B c= B
by XBOOLE_1:17;
assume
not A /\ B is empty
; A c= B
then consider a being set such that
A2:
a in A /\ B
by XBOOLE_0:def 1;
A3:
ex s being Element of A st A = {s}
by Def1;
A /\ B c= A
by XBOOLE_1:17;
then
{a} c= A
by A2, ZFMISC_1:37;
then
{a} = A
by A3, ZFMISC_1:24;
hence
A c= B
by A2, A1, ZFMISC_1:37; verum