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I have been going through Simmons' book. Class is defined as the set of sets. Given this definition, I have the following claim: $$ \text{If} \left\{ A_i \right\}\text{ and } \left\{ B_j \right\} \text{ are two classes of sets such that } \left\{ A_i \right\}\subseteq \left\{ B_j \right\}, \text{ show that } \cap_j B_{j}\subseteq\cap_i A_i $$

Though it seems simple, I could not figure out the proof. I am not sure, but I might not understand the definition completely. In any case, any help would be greatly appreciated. Thanks!

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It is not so pertinent whether the $\\{A_i\\}$ or whatever are classes. Here's a hint: $x$ is in $\cap_j B_j$ iff $x\in B_j$ for each $j$. Try unravelling the definition of $\subseteq$ as well. – tci May 5 at 23:36
If you intersect all the sets from the second class so (by hypothesis) you also intersect all the sets from the first class. Then the result should be contained in the intersection $\cap_iA_i$. – Sigur May 5 at 23:36

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Let $x\in\bigcap_j B_j$ and also pick an $i$. Then, by hypothesis, $A_i\in\{B_j\}_j$, that is, there is a $j$ such that $A_i=B_j$. So, as $x\in B_j$, we have $x\in A_i$. Since $i$ was aribtrary, $x\in \bigcap_i A_i$. -QED-

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