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import TestGame.Metadata
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import Mathlib.Data.Set.Basic
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Game "TestGame"
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World "SetTheory"
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Level 11
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Title "Strikte Teilmenge"
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Introduction
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"
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Strikte Teilmengen sind in Lean eher selten, aber wir schauen sie hier
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trotzdem kurz an : `A ⊂ B` (`\\ssub`) bedeutet `(A ⊆ B) ∧ (¬B ⊆ A)`.
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Entsprechend, kann man die gleichen Methoden wie beim UND benützen
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(`rcases`/`constructor`).
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Zudem kann man mit `rw [ssubset_def]` explizit die Definition einsetzen.
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Note: `rw [subset_def]` macht das gleiche für `⊆`.
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"
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--open Set
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Statement
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""
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(A B : Set ℕ) (h : A ⊂ B) : ∃ x, x ∈ B \ A := by
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cases' h with h₁ h₂
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rw [Set.subset_def] at h₂
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rw [not_forall] at h₂
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cases' h₂ with x hx
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use x
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rw [not_imp] at hx
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rw [Set.mem_diff]
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exact hx
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Tactics constructor intro rw assumption rcases simp tauto trivial
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Lemmas Subset.antisymm_iff empty_subset
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