|
431
|
1 <html>
|
|
|
2 <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=UTF-8">
|
|
|
3
|
|
|
4 <pre>
|
|
|
5 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
|
|
|
6 field
|
|
|
7 ZFSet : Set n
|
|
|
8 _∋_ : ( A x : ZFSet ) → Set m
|
|
|
9 _≈_ : ( A B : ZFSet ) → Set m
|
|
|
10 ∅ : ZFSet
|
|
|
11 _,_ : ( A B : ZFSet ) → ZFSet
|
|
|
12 Union : ( A : ZFSet ) → ZFSet
|
|
|
13 Power : ( A : ZFSet ) → ZFSet
|
|
|
14 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet
|
|
|
15 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet
|
|
|
16 infinite : ZFSet
|
|
|
17 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite
|
|
|
18 record IsZF {n m : Level }
|
|
|
19 (ZFSet : Set n)
|
|
|
20 (_∋_ : ( A x : ZFSet ) → Set m)
|
|
|
21 (_≈_ : Rel ZFSet m)
|
|
|
22 (∅ : ZFSet)
|
|
|
23 (_,_ : ( A B : ZFSet ) → ZFSet)
|
|
|
24 (Union : ( A : ZFSet ) → ZFSet)
|
|
|
25 (Power : ( A : ZFSet ) → ZFSet)
|
|
|
26 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet )
|
|
|
27 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet )
|
|
|
28 (infinite : ZFSet)
|
|
|
29 : Set (suc (n ⊔ m)) where
|
|
|
30 field
|
|
|
31 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_
|
|
|
32 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
33 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
34 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
|
35 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
36 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
|
37 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x}
|
|
|
38 power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t
|
|
|
39 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
|
|
|
40 regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
|
|
|
41 minimal : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet
|
|
|
42 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimal x not ∈ x ∧ ( minimal x not ∩ x ≈ ∅ ) )
|
|
|
43 ε-induction : { ψ : ZFSet → Set m}
|
|
|
44 → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x )
|
|
|
45 → (x : ZFSet ) → ψ x
|
|
|
46 infinity∅ : ∅ ∈ infinite
|
|
|
47 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite
|
|
|
48 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
|
49 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
|
50 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
|
51 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
|
52 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
|
53 </div>
|
|
|
54 <hr/> Shinji KONO <kono@ie.u-ryukyu.ac.jp> / Fri Jan 10 12:24:29 2020
|
|
|
55 </body></html>
|
