has_diff — Mathlib

English

If i ∈ S, then π i ⊆ piiUnionInter π S. This is a direct corollary of the singleton-left representation and monotonicity.

Русский

Если i ∈ S, то π i ⊆ piiUnionInter π S.

LaTeX

$$$\mathrm{Subset}(π i, piiUnionInter(π,S))$$$

Lean4

theorem has_diff {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : s₂ ⊆ s₁) : d.Has (s₁ \ s₂) :=
  by
  apply d.has_compl_iff.1
  simp only [diff_eq, compl_inter, compl_compl]
  exact d.has_union (d.has_compl h₁) h₂ (disjoint_compl_left.mono_right h)

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