Intersection Subset

From ProofWiki

Jump to: navigation, search

Theorem

The intersection of two sets is a subset of each:

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

$\displaystyle \forall T \in \mathbb S: \bigcap \mathbb S \subseteq T$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \in S \cap T$	$\implies$	$\displaystyle x \in S \land x \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Intersection
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle x \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Simplification
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle S \cap T \subseteq S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Subset

Similarly for $T$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \in S \cap T\)	\(\implies\)	\(\displaystyle x \in S \land x \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Intersection
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Simplification
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle S \cap T \subseteq S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Subset