Lemma 3.18. Assuming that:

• $G$ is $\mathbb{P}$-generic over $M$

• $E\subseteq \mathbb{P}$

• $E\in M$

• (i) If $E$ is dense below $p$, $q\le p$, then $E$ is dense below $q$.
• (ii) If $\{r:E\text{ is }\text{dense below}r\}$ is dense below $p$, then $E$ is dense below $p$
• (iii) Either $G\cap E\ne \varnothing$ or $\exists q\in G$, $\forall r\in E,r\perp q$.
• (iv) If $p\in G$, $E$ is dense below $p$, then $G\cap E\ne \varnothing$.