Intersection is Largest Subset

Theorem
$$S \cap T$$ is the largest set contained in both $$S$$ and $$T$$. That is:

$$R \subseteq S \land R \subseteq T \iff R \subseteq S \cap T$$

Proof
Let $$R \subseteq S \land R \subseteq T$$.

$$x \in R \Longrightarrow \left ({x \in S \land x \in T}\right)$$ (Definition of Subset)

$$\Longrightarrow x \in S \cap T$$ (Definition of Intersection)

$$\Longrightarrow R \subseteq S \cap T$$ (Definition of Subset)

Now let $$R \subseteq S \cap T$$.

From Intersection Subset we have $$S \cap T \subseteq S$$ and $$S \cap T \subseteq T$$.

From Subsets Transitive, it follows directly that $$R \subseteq S$$ and $$R \subseteq T$$

Thus $$R \subseteq S \land R \subseteq T \iff R \subseteq S \cap T$$.