>>778 補足
(引用開始)
集合族 A-{aξ：ξ<α} ∈S で
A-{aξ：ξ<α} を 下記に展開すると
{A,A-{a0},A-{a0,a1},A-{a0,a1,a2},・・,A-{aξ：ξ<α},・・} で、左記の集合は Sの部分集合
（明らかに、集合Aと同じ濃度）
だから、Sの部分集合の形成には、選択関数は不要（置換公理が使える）
(引用終り)

＜補足＞
１）かように、Aのべき集合全体(空集合抜き)の選択関数は不要
２）Aと同じ順序数（超限帰納）の選択関数で間に合うことを指摘しておく
３）調べると 可算集合Aを整列させるためには、従属選択公理が必要とある
（下記の独 de.wikipedia ご参照。en.wikipediaにも類似記載あり。
　即ち、”to construct a sequence using countable transfinite recursion”
　なお、Axiom of countable choice en.wikipedia は、”for every n∈N”つまり、順序数の長さでω（＝N）が限界）

(参考)
de.wikipedia.org/wiki/Axiom_der_abh%C3%A4ngigen_Auswahl
Axiom der abhängigen Auswahl
(google 英訳)
axiom of dependent choice
use
The axiom of dependent choice is a sufficient fragment of the axiom of choice to construct a sequence using countable transfinite recursion .

en.wikipedia.org/wiki/Axiom_of_dependent_choice
Axiom of dependent choice
Use
The axiom DC is the fragment of AC that is required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of those choices cannot be made independently of previous choices.

en.wikipedia.org/wiki/Axiom_of_countable_choice
Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function
A with domain (where N denotes the set of natural numbers) such that
A(n) is a non-empty set for every
n∈N, there exists a function
f with domain N such that f(n)∈A(n)
for every n∈N.

ja.wikipedia.org/wiki/%E5%8F%AF%E7%AE%97%E9%81%B8%E6%8A%9E%E5%85%AC%E7%90%86
可算選択公理