let X be set ; :: thesis: for F being non empty ordered Subset-Family of X
for G being set st G in F holds
( G = min F iff for Y being set st Y in F holds
G c= Y )
let F be non empty ordered Subset-Family of X; :: thesis: for G being set st G in F holds
( G = min F iff for Y being set st Y in F holds
G c= Y )
let G be set ; :: thesis: ( G in F implies ( G = min F iff for Y being set st Y in F holds
G c= Y ) )
assume A1: G in F ; :: thesis: ( G = min F iff for Y being set st Y in F holds
G c= Y )
thus ( G = min F implies for Y being set st Y in F holds
G c= Y ) by SETFAM_1:3; :: thesis: ( ( for Y being set st Y in F holds
G c= Y ) implies G = min F )
assume for Y being set st Y in F holds
G c= Y ; :: thesis: G = min F
then A2: G c= min F ;
min F c= G by A1, SETFAM_1:3;
hence G = min F by A2, XBOOLE_0:def 10; :: thesis: verum