I'm a little new when it comes to proof writing and was wondering if someone could help me check if my proof to the following theorem is a valid one:
Theorem: Suppose $\mathbb{F}$ and $\mathbb{G}$ are nonempty families of sets and every element of $\mathbb{F}$ is a subset of every element of $\mathbb{G}$. Then $\cup \mathbb{F} \subseteq \cap \mathbb{G}$.
Here's my proof:
Suppose $\mathbb{F}$ and $\mathbb{G}$ are nonempty families of sets, and every element of $\mathbb{F}$ is a subset of every element of $\mathbb{G}$. Now suppose $x \in \cup \mathbb{F}$. Then there is some set $A$ such that $A \in \mathbb{F}$ and $x \in A$. Since every element of $\mathbb{F}$ is a subset of every element of $\mathbb{G}$, it follows that $A \subseteq \cap \mathbb{G}$. Since $x \in A$, then $x \subseteq \cap \mathbb{G}$. But $x$ was an arbitrary element in $\cup \mathbb{F}$, so $\cup \mathbb{F} \subseteq \cap \mathbb{G}$.
Comments:
I feel uneasy about the statement “since every element of $\mathbb{F}$ is a subset of every element of $\mathbb{G}$, it follows that $A \subseteq \cap \mathbb{G}$.” Is this a valid logical deduction?
Thanks in advance for the help! Sorry if this question seems kind of simple — I just want to make sure my thought process is correct.