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How could I write the intersection of all subsets in $S$ containing $E$ in symbols?

Something like? $$\bigcap_{E\subset C}C$$

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$$\bigcap\bigl\{C\subset S:E\subset C\bigr\}$$ or $$\bigcap_{E\subset C\subset S}C$$

It's worth noting that if you actually meant the intersection of all subsets $C$ of $S$ such that $C$ contains $E$, then there's an even simpler way to write it: $$E.$$ This may, of course, be what you're trying to prove.

If the $C$ are meant to be elements of $S$, rather than subsets, simply replace $C\subset S$ with $C\in S$ in the first two versions I gave (and in that case, the intersection may not be $E$, depending on what $S$ is).

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