I'm self-learning set theory, and as an exercise, I tried to prove one of DeMorgan's laws, i.e.,
$$(S \cap T)^c=S^c \cup T^c$$
So my proof goes as follows:
Let $x \in S^c$, and $x \in T^c$. Then, $x \notin S$ and $x \notin T$. Therefore, $x$ is neither a member of $S \cup T$, nor a member of $S \cap T$. As a result, it can be seen that $(S \cap T)^c \subset S^c \cup T^c$. Being that $x$ is a member of both $(S \cap T)^c$ and $S^c \cup T^c$, it follows that $S^c \cup T^c \subset (S \cap T)^c$. Therefore, being that $(A=B) \iff (A\subset B) \land (B\subset A)$, we can conclude that $(S \cap T)^c=S^c \cup T^c$. QED. My question is, does this prove DeMorgan's law?