Maximal-antichain criterion for genericity ($|G\cap A|=1\text{ for every ground maximal antichain }A$)

A filter is generic over its ground model exactly when it meets each ground-model maximal forcing antichain in one point. Extensions of a maximal antichain form a dense set, while any ground dense set contains an antichain maximal in the whole forcing. Directedness permits at most one antichain member in a filter.