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1 module zf where
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2
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3 open import Level
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4
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5 open import logic
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6
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7 open import Relation.Nullary
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8 open import Relation.Binary hiding (_⇔_)
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9 open import Data.Empty
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10
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11 record IsZF {n m : Level }
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12 (ZFSet : Set n)
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13 (_∋_ : ( A x : ZFSet ) → Set m)
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14 (_≈_ : Rel ZFSet m)
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15 (∅ : ZFSet)
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16 (_,_ : ( A B : ZFSet ) → ZFSet)
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17 (Union : ( A : ZFSet ) → ZFSet)
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18 (Power : ( A : ZFSet ) → ZFSet)
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19 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet )
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20 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet )
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21 (infinite : ZFSet)
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22 : Set (suc (n ⊔ suc m)) where
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23 field
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24 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_
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25 -- ∀ x ∀ y ∃ z ∀ t ( z ∋ t → t ≈ x ∨ t ≈ y)
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26 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
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27 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
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28 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
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29 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
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30 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
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31 _∈_ : ( A B : ZFSet ) → Set m
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32 A ∈ B = B ∋ A
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33 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m
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34 _⊆_ A B {x} = A ∋ x → B ∋ x
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35 _∩_ : ( A B : ZFSet ) → ZFSet
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36 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) )
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37 _∪_ : ( A B : ZFSet ) → ZFSet
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38 A ∪ B = Union (A , B)
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39 ｛_｝ : ZFSet → ZFSet
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40 ｛ x ｝ = ( x , x )
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41 infixr 200 _∈_
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42 infixr 230 _∩_ _∪_
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43 infixr 220 _⊆_
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44 field
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45 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
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46 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
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1095
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47 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → A ∋ x -- _⊆_ t A {x}
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48 power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t
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49 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
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50 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
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51 -- regularity without minimum
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437
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52 ε-induction : { ψ : ZFSet → Set (suc m Level.⊔ n)}
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53 → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x )
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54 → (x : ZFSet ) → ψ x
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55 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
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56 infinity∅ : ∅ ∈ infinite
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57 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite
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58 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
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59 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
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60 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
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61 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
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62
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63 record ZF {n m : Level } : Set (suc (n ⊔ suc m)) where
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64 infixr 210 _,_
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65 infixl 200 _∋_
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66 infixr 220 _≈_
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67 field
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68 ZFSet : Set n
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69 _∋_ : ( A x : ZFSet ) → Set m
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70 _≈_ : ( A B : ZFSet ) → Set m
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71 -- ZF Set constructor
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72 ∅ : ZFSet
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73 _,_ : ( A B : ZFSet ) → ZFSet
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74 Union : ( A : ZFSet ) → ZFSet
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75 Power : ( A : ZFSet ) → ZFSet
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76 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet
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77 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet
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78 infinite : ZFSet
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79 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite
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80
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