Is my set theory notation correct?

Question

Is my set theory notation correct?

(1) Every element of $A$ is in $B$

Set theory notation: $\forall x \in A, x \in B$

(2) For every element of set $A$, $P$ is true

Set theory notation: $\forall x \in A$, $P$ is true

Answer 1

Well, no.

(1) is $\forall x(x\in A\Rightarrow x\in B)$, (2) is $\forall x(x\in A\Rightarrow P(x))$, where $P=P(x)$ is a predicate in the free variable $x$.