α : Type u_1
s : Set α
t : Set Bool
⊢ ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = univ ∨
    ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = s ∨
      ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = sᶜ ∨
        ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = ∅
α : Type u_1
s : Set α
t : Set Bool
⊢ boolIndicator s ⁻¹' t = univ ∨ boolIndicator s ⁻¹' t = s ∨ boolIndicator s ⁻¹' t = sᶜ ∨ boolIndicator s ⁻¹' t = ∅
simp only [preimage_boolIndicator_eq_union]
no goals
α : Type u_1
s : Set α
t : Set Bool
⊢ ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = univ ∨
    ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = s ∨
      ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = sᶜ ∨
        ((if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅) = ∅
split_ifs <;> simp [s.union_compl_self]