[FOM] finite choice question

[Stephen Fenner]

This is a basic question about ZF set theory.
It's not hard to see that the following is true:

METATHEOREM: For any fixed natural number n, the sentence, "For any 
sequence <X1,...,Xn> of pairwise disjoint, nonempty sets, there is a set C 
such that (C intersect Xi) is a singleton for each i in {1,...,n}" is a 
theorem of ZF.

But is the following a theorem of ZF?

"For any natural number n and any sequence <X1,...,Xn> of pairwise 
disjoint, nonempty sets, there is a set C such that (C intersect Xi) is a 
singleton for each i in {1,...,n}."

Note that, unlike other finite versions of AC, the cardinalities of the Xi 
are unrestricted.

Steve Fenner