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import Mathlib.Data.Set.Basic
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import Mathlib.SetTheory.ZFC.Basic
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import Common.Set.OrderedPair
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2023-06-07 02:16:06 +00:00
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/-! # Enderton.Chapter_3
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Relations and Functions
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-/
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namespace Enderton.Set.Chapter_3
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/--
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If `x ∈ C` and `y ∈ C`, then `⟨x, y⟩ ∈ 𝒫 𝒫 C`.
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-/
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theorem theorem_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
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: Set.OrderedPair x y ∈ 𝒫 𝒫 C := by
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have hxs : {x} ⊆ C := Set.singleton_subset_iff.mpr hx
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have hxys : {x, y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
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exact Set.mem_mem_imp_pair_subset hxs hxys
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end Enderton.Set.Chapter_3
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