>>183 補足
>証明にべき集合を使っていないね
>単純に、どんどん集合の元を取っていって、取りつくせるって

下記proofwikiに、分かり易い証明があるね
べき集合を作って、順序数への選択関数を作る
（べき集合P(S)で、選択関数用の集合族を作ったってことかな？）
超限帰納法で、どんどん集合の元を取っていって、取りつくせるってｗ（＾＾
（ところでproofwikiなんてあるんだ。やっぱ英語情報はいいね）
https://proofwiki.org/wiki/Well-Ordering_Theorem
Well-Ordering Theorem
Contents
1 Theorem
2 Proof
2.1 Basis for the Induction
2.2 Inductive Step
3 Also known as
4 Axiom of Choice

Theorem
Every set is well-orderable.

Proof
Let S be a set.
Let P(S) be the power set of S.
By the Axiom of Choice, there is a choice function c defined on P(S)＼{Φ}.
We will use c and the Principle of Transfinite Induction to define a bijection between S and some ordinal.
Intuitively, we start by pairing c(S) with 0, and then keep extending the bijection by pairing c(S＼X) with α, where X is the set of elements already dealt with.

Basis for the Induction
α=0
Let s0=c(S).

Inductive Step
Suppose sβ has been defined for all β<α.
If S＼{sβ:β<α} is empty, we stop.
Otherwise, define:
sα:=c(S＼{sβ:β<α})
The process eventually stops, else we have defined bijections between subsets of S and arbitrarily large ordinals.
(引用終り)
以上