Understanding the use of brackets in set theory notation I would appreciate help understanding the presence and absence of brackets in this particular example, which hopefully will clarify things for me.
It is a line from a short proof by contradiction based on the Axiom of Foundation that No Set is an Element of Itself. 
Why are there brackets on some $S$'s and not on others.


Suppose $S\in S$. Let $T=\{S\}$, then $T\bigcap S=\{S\}$, etc.


Thanks

Given an arbitrary set $S$, the goal is to prove that $S$ is not an element of itself. The proposed proof proceeds by considering the set $T=\{S\}$, i.e., the set $T$ whose one and only member is the set $S$. Then $T$ is nonempty, because it has a member, namely $S$. So we can apply the axiom of regularity to conclude that $T$ has a member that has no members in common with $T$.  That member of $T$ is $S$, because $S$ is the only member of $T$. So $S$ has no members in common with $T$. Tbat is, no member of $T$ can also be a member of $S$. Well, $S$ is a member of $T$, so $S$ cannot be a member of $S$, q.e.d.
