Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty
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Theorem
Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.
Let:
- $\forall \beta \in I: A_\beta \subseteq B_\beta$
Then:
- $\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O \implies \bigcap_{\alpha \mathop \in I} A_\alpha = \O$
Proof
Let $\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O$.
From Set Intersection Preserves Subsets/Families of Sets:
- $\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha = \O$
From Subset of Empty Set:
- $\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$
$\blacksquare$