Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff zf.agda @ 116:47541e86c6ac
axiom of selection
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 26 Jun 2019 08:05:58 +0900 |
| parents | 277c2f3b8acb |
| children | 0c2cbf37e002 |
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--- a/zf.agda Tue Jun 25 22:47:17 2019 +0900 +++ b/zf.agda Wed Jun 26 08:05:58 2019 +0900 @@ -15,7 +15,7 @@ case1 : A → A ∨ B case2 : B → A ∨ B -_⇔_ : {n : Level } → ( A B : Set n ) → Set n +_⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m) _⇔_ A B = ( A → B ) ∧ ( B → A ) open import Relation.Nullary @@ -74,7 +74,7 @@ -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( X x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite - selection : ∀ { X : ZFSet } → { ψ : (x : ZFSet ) → Set m } → ∀ { y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) + selection : ∀ { X : ZFSet } → { ψ : (x : ZFSet ) → Set m } → ∀ { y : ZFSet } → (((y : ZFSet) → y ∈ X → ψ y ) ∧ ( y ∈ X ) ) ⇔ (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 ) → ( ψ x ∈ Replace X ψ ) -- -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ]