Set is Subset of Intersection of Supersets/General Result
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Theorem
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:
- $\ds X \subseteq \bigcap_{i \mathop \in I} S_i$
where $\ds \bigcap_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.
Proof
Let $X \subseteq S_i$ for all $i \in I$.
Then:
| \(\ds x\) | \(\in\) | \(\ds X\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall i \in I: \, \) | \(\ds x\) | \(\in\) | \(\ds S_i\) | Definition of Subset | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall i \in I: \, \) | \(\ds x\) | \(\in\) | \(\ds \bigcap_{i \mathop \in I} S_i\) | Definition of Intersection of Family | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds X\) | \(\subseteq\) | \(\ds \bigcap_{i \mathop \in I} S_i\) | Definition of Subset |
$\blacksquare$
Sources
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets: Exercise $1.4.4 \ \text{(ii)}$