Subset Relation is Transitive

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Theorem

The relation "is a subset of" is transitive:

$\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$


Proof

\(\displaystyle \) \(\) \(\displaystyle \paren {R \subseteq S} \land \paren {S \subseteq T}\) $\quad$ $\quad$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \in R \implies x \in S} \land \paren {x \in S \implies x \in T}\) $\quad$ Definition of Subset $\quad$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \in R \implies x \in T}\) $\quad$ Hypothetical Syllogism $\quad$
\(\displaystyle \) \(\leadsto\) \(\displaystyle R \subseteq T\) $\quad$ Definition of Subset $\quad$

$\blacksquare$


Sources