Intersection is Commutative/Family of Sets/Proof 1

Proof
We have that both $\displaystyle \bigcap_{j \mathop \in J} S_j$ and $\displaystyle \bigcap_{k \mathop \in \relcomp I J} S_k$ are sets.

Hence by Intersection is Commutative we have:
 * $\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j$

It remains to be demonstrated that $\displaystyle \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k$.

So:

That is:
 * $\displaystyle x \in \bigcap_{i \mathop \in I} S_i \iff x \in \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k$

The result follows by definition of set equality.