simp_rw [subset_iff, mem_posDiffs, mem_inter]
case refine'_1
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
h𝒜 : IsLowerSet ↑𝒜
hℬ : IsUpperSet ↑ℬ
⊢ 𝒜 \₊ ℬ ⊆ 𝒜 ∩ ℬ \₊ (𝒜 ∩ ℬ)
case refine'_1
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
h𝒜 : IsLowerSet ↑𝒜
hℬ : IsUpperSet ↑ℬ
⊢ ∀ ⦃x : Finset α⦄,
    (∃ b ∈ 𝒜, ∃ c ∈ ℬ, c ≤ b ∧ b \ c = x) → ∃ b, (b ∈ 𝒜 ∧ b ∈ ℬ) ∧ ∃ c, (c ∈ 𝒜 ∧ c ∈ ℬ) ∧ c ≤ b ∧ b \ c = x