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import Adam.Metadata
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import Mathlib.Data.Set.Basic
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Game "Adam"
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World "SetTheory"
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Level 9
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Title "Komplement"
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Introduction
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"
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Das Komplement einer Menge wird als `Aᶜ` (`\\^c`) geschrieben. Wichtige Lemmas
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sind `not_mem_compl_iff` und `compl_eq_univ_diff`.
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"
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open Set
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#check not_mem_compl_iff
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#check compl_eq_univ_diff
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Statement
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""
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(A : Set ℕ) (h : Aᶜ ⊆ A) : A = univ := by
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apply Subset.antisymm
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simp only [subset_univ]
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intros x hx
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by_cases h4 : x ∈ Aᶜ
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exact mem_of_subset_of_mem h h4
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rw [←not_mem_compl_iff]
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exact h4
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NewTactic constructor intro rw assumption rcases simp tauto trivial
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