Essential disjointness of small subfamilies ($A_{\eta }^{\ast }=A_{\eta }\setminus {\bigcup }_{\nu <\eta }{A}_{\nu }$)

Any at-most-$\kappa$ subfamily of an almost disjoint family on a regular cardinal becomes pairwise disjoint after deleting fewer than $\kappa$ points from each member. Enumerate it in length at most $\kappa$ and apply the displayed trimming. Each removed part is a union of fewer than $\kappa$ small intersections, so regular cardinal arithmetic bounds it below $\kappa$.