Maximal-antichain criterion for genericity
= Maximal-antichain criterion for genericity
{title2=$|G\cap A|=1\text{ for every ground maximal antichain }A$}
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.