Understanding why x ∈ ∩∅ is true for every value of x In How to Prove It: A Structured Approach, the author mentions:

... if F = ∅, then the statement x ∈ ∩F will be true no matter what x is.

If I analyse the logical structure of x ∈ ∩∅:
x ∈ ∩∅
x ∈ {y | ∀A(A ∈ ∅ → y ∈ A)}
∀A(A ∈ ∅ → x ∈ A)
∀A(F → x ∈ A) Because ∅ has no members
∀A(T ∨ x ∈ A) Conditional law
∀A(x ∈ A)     Tautology law

My conclusion supports what the author says, however this doesn't makes sense to me.
∩F is supposed to represent the intersection of all the elements of F, but if F is ∅, then as far as I understand F will not have any element, leading ∩F to some undefined state as far as I can think of.
How can then x ∈ ∩∅ be true for every value of x?

Probably this would help a bit.
$\bigcap\varnothing$ should be the identity element of intersection operation. Now, think of the case, when all sets involved are subsets of some universe set $U$. In this case, the set $U$ clearly is the identity element for intersection (since if $A \subset U$ then $A \cap U = A$).
So, in a general case, $\bigcap\varnothing$ should be equal to the set of all sets (which does not exist in ZF, by the way).
