α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ card (filter (fun s => a ∈ s) 𝒜) + card (filter (fun s => ¬a ∈ s) 𝒜) = card 𝒜

α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ Set.InjOn (fun s => erase s a) ↑(filter (fun s => a ∈ s) 𝒜)
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ card (memberSubfamily a 𝒜) + card (nonMemberSubfamily a 𝒜) = card 𝒜
rw [memberSubfamily, nonMemberSubfamily, card_image_of_injOn]
no goals
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ card (filter (fun s => a ∈ s) 𝒜) + card (filter (fun s => ¬a ∈ s) 𝒜) = card 𝒜
conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun s => (a ∈ s))]
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ ↑(filter (fun s => a ∈ s) 𝒜) ⊆ {s | a ∈ s}
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ Set.InjOn (fun s => erase s a) ↑(filter (fun s => a ∈ s) 𝒜)
apply (erase_injOn' _).mono
no goals
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 : Finset (Finset α)
⊢ ↑(filter (fun s => a ∈ s) 𝒜) ⊆ {s | a ∈ s}
simp