simp only [Finset.mem_filter, Finset.mem_Icc, Nat.ceil_le, Finset.mem_union, and_imp]
x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : 1 ≤ z
p : ℕ
⊢ p ∈ Finset.filter Nat.Prime (Finset.Icc ⌈x⌉₊ ⌊x + y⌋₊) →
    p ∈ Finset.filter (fun d => ∀ (p : ℕ), p.Prime → ↑p ≤ z → ¬p ∣ d) (Finset.Icc ⌈x⌉₊ ⌊x + y⌋₊) ∪ Finset.Icc 1 ⌊z⌋₊
x y z : ℝ
hx : 0 < x
hy : 0 < y
hz : 1 ≤ z
p : ℕ
⊢ x ≤ ↑p →
    p ≤ ⌊x + y⌋₊ → p.Prime → ((x ≤ ↑p ∧ p ≤ ⌊x + y⌋₊) ∧ ∀ (p_1 : ℕ), p_1.Prime → ↑p_1 ≤ z → ¬p_1 ∣ p) ∨ 1 ≤ p ∧ p ≤ ⌊z⌋₊