# Set is Subset of Intersection of Supersets

## 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

$\blacksquare$