Set is Subset of Intersection of Supersets

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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:

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


Set of Sets

Let $T$ be a set.

Let $\mathbb S$ be a set of sets.

Suppose that for each $S \in \mathbb S$, $T \subseteq S$.


Then:

$T \subseteq \displaystyle \bigcap \mathbb S$


General Result

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

Let $X$ be a set such that:

$\forall i \in I: X \subseteq S_i$


Then:

$X \subseteq \bigcup_{i \mathop \in I} S_i$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.


Proof 1

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


Proof 2

\(\displaystyle S\) \(\subseteq\) \(\displaystyle T_1\)
\(\, \displaystyle \land \, \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle T_2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle S \cap S\) \(\subseteq\) \(\displaystyle S \cap T_2\) Set Intersection Preserves Subsets
\(\displaystyle \leadsto \ \ \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle T_1 \cap T_2\) Intersection is Idempotent

$\blacksquare$


Also see


Sources