Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty

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:

$\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$

$\blacksquare$