How Can Intersection of Indexed Family Sets Belong to Their Power Sets?

[nike5]

Homework Statement

Suppose {A_i| i \in I} is an indexed family of sets and I does
equal an empty set. Prove that \bigcap i \in I Ai
\in \bigcap i\in I P(Ai ) and P(Ai) is the
power set of Ai

Homework Equations

none

The Attempt at a Solution

Suppose x \in {A_i| i \in I}. Let i be an arbitrary element of
I where x \in Ai . Then let y be an arbitrary element of x. Since x
is an element of Ai and y \in x it follows that ...

maybe i want to show that \bigcap i \in I Ai \subseteq \bigcap i \in I Ai and then
I could say that \bigcap i \in I Ai \in \bigcap i\in I P(Ai )

 

[konthelion]

Let \left\{ A_{i} \right\}_{i \in I} be your indexed set of family.

Do you mean this \bigcap_{i=1} A_i = \left\{ x : \forall i \in I: x \in A_i \right\}?

 

[nike5]

Yes sry about the horrible looking symbols

 

[HallsofIvy]

Homework Statement

Suppose {A_i| i \in I} is an indexed family of sets and I does
equal an empty set.

Did you mean "does not equal and empty set"?

 

[nike5]

I \neq \oslash is what I meant