>>777
＞∀i∈{1,2,...,n},S_i={0}　とすれば
＞∀i∈{1,2,...,n},0∈S_i、(0,0,...,0)∈π[i=1,n]S_i　だが

だろ？

それ「有限個の要素による集合族」に対する選択「公理」の否定になってないな

いっとくけど
「個々の要素が有限集合の（要素数が無限個でもよい）集合族の選択公理」
なんか論じてないぞｗ

Restriction to finite sets
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by mathematical induction.[6] In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.