Big Union Big Intersection Proof

I'm having trouble working out how to prove the following proposition:

If A and B are non-empty and ∀X∈A.∀Y∈B. X⊆Y then ∐A ⊆ ∩B.

I know that ∐A = {x|∃X ∈A. x∈X} and ∩B = {y|∀Y∈B. y∈Y} and to show the subset relation holds I need to show that x∈∐A ⇒ x∈∩B. I feel like I understand why the proposition has to hold intuitively but can't seem to come up with a proof.

Thanks in advance.

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