exact hx3.trans (mem_Ioc.1 (mem_filter.1 $ min'_mem _ _).1).2
case pos
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
i : ℕ
hi : i ∈ A
x : ℕ
hx1 : x ∈ A + B ∧ i < x
hx2 : x ∈ A + B
h : (filter (fun x => x ∈ A) (Ioc i n)).Nonempty
hx3 : x ≤ (filter (fun x => x ∈ A) (Ioc i n)).min' ⋯
⊢ x ≤ n
no goals