Simple Cardinality Proof

So I'm trying to perform a simple proof using cardinalities. It looks like:

⟦(A::nat set) ∩ B = {}⟧ ⟹ (card (A ∪ B) = card A + card B)

Which seems to makes sense, but for some reason blast hangs, the rest of the provers fail to apply, and sledgehammer times out. Is there a gap in what I think I know about cardinalities? If not, how can I prove this lemma?

Thanks in advance!

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