disjoint_powerset_insert — Mathlib

English

If a ∉ s then the two families 𝒫(s) and insert a '' 𝒫(s) are disjoint as sets of subsets.

Русский

Если a ∉ s, то множества 𝒫(s) и insert a '' 𝒫(s) несовместны (дисjoint).

LaTeX

$$$\\operatorname{Disjoint}(\\mathcal{P}(s), \\operatorname{insert}(a)''\\mathcal{P}(s))$ if $a \\notin s$$$

Lean4

theorem disjoint_powerset_insert {s : Set α} {a : α} (h : a ∉ s) : Disjoint (𝒫 s) (insert a '' 𝒫 s) :=
  by
  rw [Set.disjoint_iff_forall_ne]
  refine fun u u_mem v v_mem ↦ (ne_of_mem_of_not_mem' ?_ (Set.notMem_subset (Set.subset_of_mem_powerset u_mem) h)).symm
  simp only [mem_powerset_iff, mem_image] at v_mem
  obtain ⟨_, _, eq⟩ := v_mem
  simp [← eq]

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