# Set theory notation for intersection

I've started reading Probability by Blikzstein and Wang and run into the following formula:

$$B = \bigcap_{j=1}^{10} A_{j}$$

I wasn't able to find a definition of the notation in the book, or on Wikipedia:

I think I can guess what the notation means, but I was hoping to find a formal definition.

I have seen a similar question here: Set Theory Notation Crises, however that question does not have the digits above the intersection symbol.

In set theory given a class term $$X$$ its intersection is defined as $$\bigcap X := \{x \mid \forall y \in X, x \in y\}.$$ Now, note that we can equivalently write $$\bigcap_{j = 1}^{10} A_j = \bigcap \{A_j \mid \forall j \in \{1, \ldots, 10\}\}.$$ Therefore $$\bigcap_{j = 1}^{10} A_j = \{x \mid \forall j \in \{1, \ldots, 10\}, x \in A_j\}.$$ Check out 'Introduction to set theory' by Hrbacek and Jech for further info.

It's a short-hand notation for $$B= A_1\cap A_2\cap \ldots \cap A_{10}$$.