Countable set question

Hello everyone, I am preparing for a final and would like some help.

Suppose:
A_0 >= A_1 >= A_2 >= ... >= A_alpha >= ... (alpha < w_1) is a sequence of disjoint sets. (A >= B reads as "B is a subset of A").

How does one prove or disprove that there must be an alpha < w_1 with A_alpha a countable set?
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