Take a look at the image below:
The blue part is of Paul.R.Halmos naive set theory.
I looked for the formal statement of the axiom and i found 2 different statements on net!
The red and the green one!
Certainly they are different. In fact the red is a part of the green as you know.
I myself think the red is true and this also corresponds to Halmos assertion.
Because if the green were true, then the brown weren't so!(Since every element of B can be found in at least one of the sets contained in A!)
What do you think?!
(Sorry for different letters in different assertions-It's just a collection of copies and so i didn't change the letters!)