Is it notationally appropriate to use $\in$ and $\subseteq$ in the quantificational part of a predicate logic statement?

Question

1. In this question it was said in a comment that it is not good style to use $\in$ in the quantificational part of a predicate logic formula.
   Is this so? I have actually seen this quite often in definitions (like $\forall x \in \mathbb{R}:\ ...$) and find this by far more convenient than definining additonal predicates for sets that already have an unambiguous set symbol (like $\mathbb{R}$) and then spelling out the set membership by an additional predicate $P(x) \land \ ...$ (for $\exists x$) or $P(x) \to \ ...$ (for $\forall x$) respectively.
   In some statements (usually translations of natural language sentences into predicate logic such as "All bachelors are not married") this predicate version instead of $\in$ notaton might make sense, but is it in general not good style to write $\exists x \in P : R(x)$ but only $\exists x : P(x) \land R(X)$?
2. Similarly, is it permissible to use the $\subseteq$ in the quantificational part of a statement?
   E.g. if I wanted to say "for all subsets $B$ of this particular set $A$", with the $\in$-relation, I'd have to make the rather complicated assertion $\forall B \in \mathbb{P}(A)$ by saying "for all sets that are an element of $A$'s power set".
   Could I just use $\forall B \subseteq A :\ ...$ instead? It looks weird to me, but I see no reason not to use it if $\in$ is allowed as well.
   In case neither $\in$ or $\subseteq$ is allowed, how would I formalize this statement - would I need to define a predicate "is a subset of $A$"?

Answer 1

Technically, it's not appropriate, in the sense that the syntax of first-order logic doesn't allow for it.

That said, it is indeed efficient. More than that, it can be viewed as an abbreviation for perfectly correct syntax: as you observe, we can just replace "$\forall x\in A(...)$" with "$\forall x(x\in A\rightarrow . . . )$," and replace "$\exists x\in A(...)$" with "$\exists x(x\in A\wedge ...)$". (A similar trick works for "$\subseteq$".)

Whether this matters comes down to what context you are in. There are certain contexts where it's important to pay attention to the precise form of a first-order sentence, and there abbreviations can mislead you as to the true complexity of the formulas involved. However, for the vast majority of purposes there is no problem.

Answer 2

In pretty much all of everyday mathematical writing, not only is it good notation to use restricted quantifiers; it is actually desirable to do so consistently, for clarity and to avoid mistakes in writing. For example, $x$ is often used to denote a real number; but often enough, our mathematical statement applies only for positive or non-negative $x$, or perhaps $x$ represents a set, a vector, a function, or some other type of mathematical object. This restricted quantification also reflects how we naturally think. For example, consider $$(\forall\varepsilon)(\forall x)(\exists\delta)[ \delta\in\Bbb R_+^n\;\&\;((\varepsilon>0\; \&\; x\in\Bbb R^n)\implies X(x,\varepsilon,\delta))],$$where $X(x,\varepsilon,\delta)$ is some mathematical statement involving (unquantified) $x$, $\varepsilon$, and $\delta$. It starts off by introducing things which could be almost any imaginable sort of mathematical object. Only after this hyper-abstract introduction are we allowed to know in which sets these objects respectively reside. How much more natural it is to write $$(\forall\,\varepsilon>0)(\forall\,x\in\Bbb R^n)(\exists\,\delta\in\Bbb R_+^n)[X(x,\epsilon,\delta)].$$