In his Tohoku paper, section 1.5, Grothendieck states the following axioms that an abelian category might satisfy:
AB4)Infinite sums exist, and the direct sum of monomorphisms is a monomorphism.
AB5)Infinite sums exist, and the and if $A_i$ (indices in some possibly infinite set $I$) is a filtrated family of subsets of some object A in the category, and B another subset of A, then $(\sum A_i)\cap B = \sum (A_i\cap B)$
(A subset is what I am translating sous-truc as meaning... I am not sure if this is the correct English notation for this notion.)
Grothendieck states that AB5 is stronger than AB4, without proof. I cannot prove it myself; can someone enlighten me as to why this is true?