Subset Relation is Transitive

Theorem
$$\left({R \subseteq S}\right) \land \left({S \subseteq T}\right) \Longrightarrow R \subseteq T$$

Proof
$$\left({R \subseteq S}\right) \land \left({S \subseteq T}\right)$$

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

$$\Longrightarrow \left({x \in R \Longrightarrow x \in T}\right)$$ Hypothetical Syllogism

$$\Longrightarrow R \subseteq T$$ Definition of subset