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import TestGame.Metadata
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import Mathlib.Data.Set.Basic
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import Mathlib.Algebra.Parity
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import Mathlib.Tactic.Ring
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import TestGame.ToBePorted
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Game "TestGame"
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World "SetTheory2"
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Level 2
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Title "Potenzmenge"
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Introduction
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"
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Eine andere wichtige Menge ist die Potenzmenge einer Menge, welche als
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`𝒫 A` geschrieben wird (`\\powerset`). Diese ist als `{S | S ⊆ A}` definiert, also
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alle Mengen, die in $A$ enthalten sind.
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Eines der wichtigsten Lemmas ist `mem_powerset_iff: x ∈ 𝒫 s ↔ x ⊆ s` welches
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im Grunde die Definition einsetzt.
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"
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open Set
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Statement
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"" (X Y : Set ℕ):
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𝒫 X ∪ 𝒫 Y ⊆ 𝒫 (X ∪ Y) := by
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intro A hA
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rw [mem_union] at hA
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simp_rw [mem_powerset_iff] at *
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rcases hA with hA | hA
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apply subset_union_of_subset_left
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assumption
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apply subset_union_of_subset_right
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assumption
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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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