let I, G, H be set ; :: thesis: for sf being Subset-Family of I st G in sf & G misses H holds
Intersect sf misses H
let sf be Subset-Family of I; :: thesis: ( G in sf & G misses H implies Intersect sf misses H )
assume that
A1: G in sf and
A2: G misses H ; :: thesis: Intersect sf misses H
Intersect sf = meet sf by A1, SETFAM_1:def 9;
hence Intersect sf misses H by A1, A2, SETFAM_1:8; :: thesis: verum