Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/zfc.agda @ 923:85f6238a38db
use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 19 Oct 2022 08:41:42 +0900 |
parents | a5f8084b8368 |
children | 76df157f6a3f |
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module zfc where open import Level open import Relation.Binary open import Relation.Nullary open import logic record IsZFC {n m : Level } (ZFSet : Set n) (_∋_ : ( A x : ZFSet ) → Set m) (_≈_ : Rel ZFSet m) (∅ : ZFSet) (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) : Set (suc (n ⊔ suc m)) where field -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A infixr 200 _∈_ infixr 230 _∩_ _∈_ : ( A B : ZFSet ) → Set m A ∈ B = B ∋ A _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) record ZFC {n m : Level } : Set (suc (n ⊔ suc m)) where field ZFSet : Set n _∋_ : ( A x : ZFSet ) → Set m _≈_ : ( A B : ZFSet ) → Set m ∅ : ZFSet Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet isZFC : IsZFC ZFSet _∋_ _≈_ ∅ Select