TITLE: Minimal blocks for a family of finite sets
QUESTION [7 upvotes]: In this question I asked for a reference for the following lemma: 
Lemma X: For every family $\mathcal G$ of nonempty finite sets there
is a minimal "blocking set" $B$.   By a "blocking set"  $B$ I mean a set which meets each $F\in\mathcal G$, and "minimal" refers to the subset relation. 
Questions: 

What is the exact strength (or approximate strength) of this lemma in the hierarchy of consequences of AC?   Clearly it implies that for every family of finite sets there is a choice function (as the finite sets can easily be made disjoint, and for a disjoint family a minimal blocking set is the same as a 
choice function.)

In particular, is lemma X equivalent to AC for families of finite sets? Or does it at least follow from the (stronger)  ordering principle?  Or at least from the (even stronger) Prime Ideal Theorem? 
Is there a natural condition on $\mathcal G$ which is weaker than "all elements of $\mathcal G$ are finite", but still implies the conclusion of the lemma above?

REPLY [3 votes]: Dear Martin: I think you can easily extend the result for systems of infinite sets, where there is a set $A$ intersecting all sets in finitely many but at least 1 point. The latter property was investigated by Erdos and Hajnal in On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87--123, see http://www.renyi.hu/~p_erdos/1961-11.pdf 
Perhaps I also had some papers on this. Does this make any sense?