[FOM] Must every proper class contain an infinite subset?

[Johan Belinfante]

This only applies to theories with proper classes, so it has no 
relevance to ZFC, etc...
To be definite, consider this to be a question about Goedel's 1939 
theory of classes.

Question:  Must every proper class contain an infinite subset?  If so, 
does the proof
require the global axiom of choice and/or the axiom of regularity 
(Goedel's Axioms D, E)?

Writing  P(x) for the class of all subsets of  x,  FINITE for the class 
of all finite sets,
and  V  for the class of all sets, this boils down to the question 
whether the following
implication holds for all  x:

subclass(P(x), FINITE)  => member(x,  V).