Halmos states in Section 1
"The most basic property of belonging is its relation to equality, which can be formulated as follows.
Axiom. of extension. Two sets are equal if and only if they have the same elements.
With greater pretentiousness and less clarity: a set is determined by its extension.
It is valuable to understand that the axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging. One way to come to understand the point is to consider a partially analogous situation in which the analogue of the axiom of extension does not hold. Suppose, for instance, that we consider human beings instead of sets, and that, if x and A are human beings, we write $x \, \epsilon \, A$ whenever x is an ancestor of A. (The ancestors of a human being are his parents, his parents' parents, their parents, etc., etc.) The analogue of the axiom of extension would say here that if two human beings are equal, then they have the same ancestors (this is the "only if" part, and it is true), and also that if two human beings have the same ancestors, then they are equal (this is the "if" part, and it is false)."
I do follow the logic of humans, ancestors but do not get how the relation between belonging and equality is shown either by the axiom or the analogy ?
In his analogy, there is the second part also that if two human beings have the same ancestors, then they are equal which is false. Why does the axiom as stated does not mention the second part? I have read elsewhere it has something to do with logic theory in Jech.