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import TestGame.Metadata
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import Mathlib.Init.Set
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import Mathlib.Tactic.Tauto
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set_option tactic.hygienic false
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Game "TestGame"
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World "SetTheory"
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Level 3
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Title "Teilmengen"
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Introduction
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"
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Hat man zwei Mengen `(A B : Set ℕ)` kann man fragen, ob diese Teilmengen
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voneinander sind: `A ⊆ B` (`\\sub`/`\\ss`) ist die Notation für Teilmengen, die auch gleich
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sein können.
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`A ⊆ B` ist als `∀ x, x ∈ A → x ∈ B` definiert, das heisst, ein `⊆` kann immer auch mit `intro x hx`
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angegangen werden.
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Die Taktik `tauto` macht das automatisch, aber um dies zu lernen ist `tauto` hier deaktiviert.
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Benutze also `intro`:
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"
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namespace MySet
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open Set
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theorem mem_univ {A : Type _} (x : A) : x ∈ (univ : Set A) := by
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trivial
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theorem not_mem_empty {A : Type _} (x : A) : x ∉ (∅ : Set A) := by
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tauto
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Statement subset_empty_iff
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"." (A : Set ℕ) : A ⊆ univ := by
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intro h hA
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apply mem_univ -- or `trivial`.
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Tactics intro trivial apply
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-- blocked: tauto simp
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end MySet
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