set theory – U contains a+b+c elements from A u B u C where U intersect A has at least a members, etc. U can be obtained from (A, B, C) choose (a, b, c)?

I have a number of overlapping sets. I need to choose some number of elements out of each one, with no resulting duplicates.

In this case, it is easier to choose the total number of elements from the union of all sets, and check if at least a elements are in A, at least b elements in B, etc.

My intuition wants me to beleive that this is enough to guarantee that a coloring of those elements with a colored A, b colored B, etc. is always possible. I do not need to know what that coloring is, just have a proof on hand that one exists.

Example:

A={1,2,3,4,5,6,7,8} choose 3
B={2,4,6,7,9,10} choose 2
C={3,4,5,9,11,12} choose 2

U={1,2,3,4,5,6,7} has 7 elements shared with A, 4 shared with B, and 3 shared with C. (7,4,3)>=(3,2,2). And can be colored {1,2,3} bigcup {6,7} bigcup {4,5}