This is true.
To prove the same we have A \cup B = A iff B \subseteq A
Let us take 2 sets A_1,A_2 which are disjoint and because it is true for every set A_1 \cup B = A_1 so B \subseteq A_1
and A_2 \cup B = A_2 so B \subseteq A_2
So from above 2 we have
B \subseteq A_1 \cap A_2
Because A_1,A_2 are disjoint sets so we have A_1 \cap A_2= \emptyset
So B = \emptyset
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