is there a problem with the following definition? definition. let $x$ be a set. a nonempty collection $\mathcal{f}$ of nonempty subsets of $x$ is a filter if and only if 1. $a \in \mathcal{f}$ and $a \subseteq b$ implies $b \in \mathcal{f}$, and 2. $a \in \mathcal{f}$ and $b \in \mathcal{f}$ implies $a \cap b \in \mathcal{f}$.
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