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What is the meaning of a formulation like: "A iff ∩A ⊆p"

A is a set of propositions, p is a specific proposition, and the whole formulation is explicated as "There is no possible world where all members of A are true but p is not". I know there must be something going on implicitly, but I can't figure out what (am I supposed to read: "A iff ∅∩A ⊆p"? But ∅∩A is the empty set again.)

Another example is: "A iff p⊆∩A"

Any help is appreciated, I've never come across anything like it.

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In Mathematical contexts a one-place intersection operator usually applies to an indexed set of sets. Thus A, here, would not merely be a set containing simple objects, but rather A is a set of sets. Perhaps, for instance, A = { X1, X2, X3 } so that therefore ∩A = X1 ∩ X2 ∩ X3.

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  • Note the similarity to the syntax in sigma-notation. If you want to sum a bunch of numbers, you write Σaᵢ this means a₁ + a₂ + ...
    – Addem
    Apr 26, 2013 at 3:11

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