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53 lines
1.1 KiB
Plaintext
53 lines
1.1 KiB
Plaintext
import TestGame.Metadata
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import TestGame.Levels.SetTheory.L04_SubsetEmpty
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--import Mathlib.Data.Set.Basic
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import Mathlib.Init.Set
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import Mathlib.Tactic.Tauto
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import Mathlib.Tactic.PushNeg
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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 5
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Title "Nonempty"
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Introduction
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"
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Das Gegenteil von `A = ∅` ist `A ≠ ∅`, aber in Lean wird der Ausdruck `A.Nonempty` bevorzugt.
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Dieser ist dadurch existiert, dass in `A` ein Element existiert: `∃x, x ∈ A`.
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Zeige dass die beiden Ausdrücke äquivalent sind:
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"
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namespace MySet
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open Set
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theorem subset_empty_iff {A : Type _} (s : Set A) : s ⊆ ∅ ↔ s = ∅ := by
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constructor
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intro h
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rw [Subset.antisymm_iff]
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constructor
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assumption
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simp only [empty_subset]
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intro a
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rw [Subset.antisymm_iff] at a
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rcases a with ⟨h₁, h₂⟩
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assumption
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Statement eq_empty_iff_forall_not_mem
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""
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{A : Type _} (s : Set A) :
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s = ∅ ↔ ∀ x, x ∉ s := by
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rw [←subset_empty_iff]
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rfl -- This is quite a miracle :)
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NewTactics constructor intro rw assumption rcases simp tauto trivial
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NewLemmas Subset.antisymm_iff empty_subset
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end MySet
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