Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison zf.agda @ 4:c12d964a04c0

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 11 May 2019 11:10:53 +0900
parents e7990ff544bf
children d9b704508281
comparison
equal deleted inserted replaced
3:e7990ff544bf 4:c12d964a04c0
58 A ∪ B = Restrict ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) 58 A ∪ B = Restrict ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) )
59 infixr 200 _∈_ 59 infixr 200 _∈_
60 infixr 230 _∩_ _∪_ 60 infixr 230 _∩_ _∪_
61 infixr 220 _⊆_ 61 infixr 220 _⊆_
62 field 62 field
63 empty : ( x : ZFSet ) → ¬ ( ∅ ∋ x ) 63 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
64 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) 64 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
65 power→ : ( A X t : ZFSet ) → A ∋ t → ∀ {x} {y} → _⊆_ t X {x} {y} 65 power→ : ( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y}
66 power← : ( A X t : ZFSet ) → ∀ {x} {y} → _⊆_ t X {x} {y} → A ∋ t 66 power← : ( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t
67 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) 67 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
68 extentionality : ( A B z : ZFSet ) → A ∋ z ⇔ B ∋ z → A ≈ B 68 extentionality : ( A B z : ZFSet ) → A ∋ z ⇔ B ∋ z → A ≈ B
69 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) 69 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
70 smaller : ZFSet → ZFSet 70 -- smaller : ZFSet → ZFSet
71 regularity : ( x : ZFSet ) → ¬ (x ≈ ∅) → smaller x ∩ x ≈ ∅ 71 -- regularity : ( x : ZFSet ) → ¬ (x ≈ ∅) → smaller x ∩ x ≈ ∅
72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) 72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
73 infinite : ZFSet 73 infinite : ZFSet
74 infinity∅ : ∅ ∈ infinite 74 infinity∅ : ∅ ∈ infinite
75 infinity : ( x : ZFSet ) → x ∈ infinite → ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite 75 infinity : ( x : ZFSet ) → x ∈ infinite → ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite
76 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) 76 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )