Syntax and Semantics Concerning Power Sets

Let $S$ be an arbitary set and $P(S)$ be the power set of S. Then, when we say consider a subset $E$ of $P(S)$, do we mean it syntactically or semantically? In other words, do we mean $E$ is a subset of $P(S)$ and so is a collection of subsets of $S$, or do we mean $E$ is a subset of $P(S)$ and so is just one single element of $P(S)$ as all elements of $P(S)$ are originally subsets of $S$?

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