Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/zf.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 | 2b5d2072e1af |
children | 08b6aa6870d9 |
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module zf where open import Level open import logic open import Relation.Nullary open import Relation.Binary hiding (_⇔_) open import Data.Empty record IsZF {n m : Level } (ZFSet : Set n) (_∋_ : ( A x : ZFSet ) → Set m) (_≈_ : Rel ZFSet m) (∅ : ZFSet) (_,_ : ( A B : ZFSet ) → ZFSet) (Union : ( A : ZFSet ) → ZFSet) (Power : ( A : ZFSet ) → ZFSet) (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) (infinite : ZFSet) : Set (suc (n ⊔ suc m)) where field isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ -- ∀ x ∀ y ∃ z ∀ t ( z ∋ t → t ≈ x ∨ t ≈ y) pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y ) pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) _∈_ : ( A B : ZFSet ) → Set m A ∈ B = B ∋ A _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m _⊆_ A B {x} = A ∋ x → B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) _∪_ : ( A B : ZFSet ) → ZFSet A ∪ B = Union (A , B) {_} : ZFSet → ZFSet { x } = ( x , x ) infixr 200 _∈_ infixr 230 _∩_ _∪_ infixr 220 _⊆_ field empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) -- regularity without minimum ε-induction : { ψ : ZFSet → Set (suc m Level.⊔ n)} → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) → (x : ZFSet ) → ψ x -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ { x }) ∈ infinite selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) record ZF {n m : Level } : Set (suc (n ⊔ suc m)) where infixr 210 _,_ infixl 200 _∋_ infixr 220 _≈_ field ZFSet : Set n _∋_ : ( A x : ZFSet ) → Set m _≈_ : ( A B : ZFSet ) → Set m -- ZF Set constructor ∅ : ZFSet _,_ : ( A B : ZFSet ) → ZFSet Union : ( A : ZFSet ) → ZFSet Power : ( A : ZFSet ) → ZFSet Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet infinite : ZFSet isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite