The (∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (x , y)) argument should come before the lists xs/ys the proof is being applied to
(∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (x , y))
xs
ys
Doesn't sound well-scoped?
Doh of course. I must have generalised it when porting it from my own library!