# Definition >[!info] Definition > Let $\mathbb{P}$ be a poset, $\mathcal{S}$ a collection of subsets of $\mathbb{P}$, $A \in \mathcal{S}$. $q$ is *$(A,\mathbb{P})$-strongly generic* if for every $r\leq q$ there is a condition $r|A\in A$ such that if $u\in A$ and $u\leq r|A$ then $u \parallel r$. > > Let $\mathbb{P}$ be a poset $\mathbb{P}\in M\prec H_{\theta}$. $q$ is *$(M,\mathbb{P})$-strongly generic* if for every $r\leq q$ there is $r|M\in \mathbb{P}\cap M$ such that if $u\in M$ and $u\leq r|M$ then $u\parallel r$. > Equivalently: $q$ forces that $\dot{G}\cap M$ is **$V$-generic** for $\mathbb{P}\cap M$. > >[!note] Remark >$q$ is $(M,\mathbb{P})$-generic if it forces that $\dot{G}\cap M$ is **$M$-generic** for $M\cap \mathbb{P}$. A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is $\mathbb{P}$ is *strongly proper* if for every $p \in \mathbb{P}$, every sufficiently large $\lambda$ ($\lambda>\left( 2^{|\mathbb{P}|} \right)^{+}$) and every countable $M \prec\left(H_\lambda, \in,<\right)$ containing $\mathbb{P}$ and $p$, there exists a $q \leq p$ that is $(M, \mathbb{P})$-strongly-generic. # Properties # Relations ## Implied by - [[]] ## Implies - [[Proper forcing]] # Preservation