Dense-below generic meeting lemma
= Dense-below generic meeting lemma
If $p\in G$ and a ground-model <set> $D$ is <dense below a forcing condition> $p$, then the <generic filter> $G$ meets $D$. Add all <incompatible forcing conditions> with $p$ to $D$. The enlarged <set> is globally dense: a condition compatible with $p$ first has a common strengthening, then a further strengthening in $D$. Genericity meets the enlargement, and directedness excludes its incompatible part.