consider A, B being set such that
A1: ( A in F & B in F & ( for Y being set holds
( Y in F iff ( A c= Y & Y c= B ) ) ) ) by DefOr;
K: for Z1 being set st Z1 in F holds
Z1 c= B by A1;
A2: A c= union F by A1, ZFMISC_1:92;
union F c= B by K, ZFMISC_1:94;
hence max F is Element of F by A1, A2; :: thesis: verum