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import TestGame.Metadata
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import TestGame.Levels.SetTheory.L05_Empty
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import Mathlib.Data.Set.Basic
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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 6
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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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open Set
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Statement nonempty_iff_ne_empty
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""
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{A : Type _} (s : Set A) :
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s.Nonempty ↔ s ≠ ∅ := by
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rw [Set.Nonempty]
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rw [ne_eq, eq_empty_iff_forall_not_mem]
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push_neg
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rfl
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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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