Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison zf.agda @ 65:164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
lemma ox = TransFinite {suc n} {λ ox → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} } { }0 { }1 { }2 ox
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 30 May 2019 01:02:47 +0900 |
| parents | 33fb8228ace9 |
| children | 93abc0133b8a |
comparison
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| 64:87df00599a0e | 65:164ad5a703d8 |
|---|---|
| 66 field | 66 field |
| 67 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) | 67 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 68 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) | 68 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 69 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → _⊆_ t A {x} | 69 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → _⊆_ t A {x} |
| 70 power← : ∀( A t : ZFSet ) → ∀ {x} → _⊆_ t A {x} → Power A ∋ t | 70 power← : ∀( A t : ZFSet ) → ∀ {x} → _⊆_ t A {x} → Power A ∋ t |
| 71 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) | 71 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 72 extentionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B | 72 extensionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B |
| 73 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) | 73 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 74 minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet | 74 minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
| 75 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | 75 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) |
| 76 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) | 76 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 77 infinity∅ : ∅ ∈ infinite | 77 infinity∅ : ∅ ∈ infinite |