Hi, I need to prove de Morgan's laws for generalised union and intersections, I came up with a proof but I don't know if it's works with what I know.

Here is what I know:

Given A, a set and B such that $\displaystyle B \subseteq A$ then $\displaystyle B^{c}=A-B$.

$\displaystyle \mathcal{A}$ is a collection of sets.

$\displaystyle \bigcup_{A \in \mathcal{A}}A=\{ x| \exits A \in \mathcal {A} \text{ such that } x \in A \}$.

$\displaystyle \bigcap_{A \in \mathcal{A}}A= \{x| x \in A, \forall A \in \mathcal A\}$.

proof:

$\displaystyle \text{1. }(\bigcup_{A \in \mathcal{A}}A)^{c}=\{x| x \notin A, \forall A \in \mathcal{A}\}=\{x|x \in A^{c}, \forall A \in \mathcal{A}\}=\bigcap_{A \in \mathcal{A}}A^{c}$.

$\displaystyle \text{2. }(\bigcap_{A \in \mathcal{A}}A)^{c}=\{x| \exits A \in \mathcal{A} \text{ such that } x \notin A\}= \{ x| \exists A \in \mathcal{A} \text{ such that } x \in A^{c}\}=\bigcup_{A \in \mathcal{A}}A^c$.

Thanks in advance!.