Set is Subset of Intersection of Supersets/Proof 1

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Theorem

Let $S$, $T_1$ and $T_2$ be sets.

Let $S$ be a subset of both $T_1$ and $T_2$.


Then:

$S \subseteq T_1 \cap T_2$


That is:

$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$


Proof

Let $S \subseteq T_1 \land S \subseteq T_2$.


Then:

\(\displaystyle x \in S\) \(\leadsto\) \(\displaystyle x \in T_1 \land x \in T_2\) Definition of Subset
\(\displaystyle \) \(\leadsto\) \(\displaystyle x \in T_1 \cap T_2\) Definition of Set Intersection
\(\displaystyle \) \(\leadsto\) \(\displaystyle S \subseteq T_1 \cap T_2\) Definition of Subset


Sources