@@ -23,17 +23,18 @@ def isSetPi₁' (A: U₁) (B: A → U) (h: Π (x: A), isSet (B x)) : isSet₁ (
2323
2424def Ω := U → 𝟐
2525def ℙ (X: U₁) := X → Ω
26- def isSet-ℙ (X: U₁) : isSet₁ (ℙ X)
26+
27+ def isSet-ℙ (X: U₁)
28+ : isSet₁ (ℙ X)
2729 := isSetPi₁ X (λ (x: X), Ω) (λ (x: X), isSetPi₁' U (λ (_: U), 𝟐) (λ (_: U), boolset))
30+
2831axiom specify (X: U₁) : (X → Ω) → ℙ X
32+
2933def ∅ (X: U₁) : ℙ X := \ (_: X) (_: U), false
3034def total (X: U₁) : ℙ X := \ (_: X) (_: U), true
3135def ∈ (X: U₁) (el: X) (set: ℙ X) : U₁ := Path₁ (U → 𝟐) (set el) (\(_: U), true)
3236def ∉ (X: U₁) (el: X) (set: ℙ X) : U₁ := Path₁ (U → 𝟐) (set el) (\(_: U), false)
3337def ⊆ (X: U₁) (A B: ℙ X) := Π (x: X), (∈ X x A) × (∈ X x B)
34- def ∁ (X: U₁) : ℙ X → ℙ X -- not (complement)
35- := λ (h: ℙ X), λ (x: X) (Y: U), not (h x Y)
36- def ∪ (X: U₁) : ℙ X → ℙ X → ℙ X -- or (union)
37- := λ (h1: ℙ X) (h2: ℙ X), λ (x: X) (Y: U), or (h1 x Y) (h2 x Y)
38- def ∩ (X: U₁) : ℙ X → ℙ X → ℙ X -- and (intersection)
39- := λ (h1: ℙ X) (h2: ℙ X), λ (x: X) (Y: U), and (h1 x Y) (h2 x Y)
38+ def ∁ (X: U₁) : ℙ X → ℙ X := λ (h: ℙ X), λ (x: X) (Y: U), not (h x Y)
39+ def ∪ (X: U₁) : ℙ X → ℙ X → ℙ X := λ (h1: ℙ X) (h2: ℙ X), λ (x: X) (Y: U), or (h1 x Y) (h2 x Y)
40+ def ∩ (X: U₁) : ℙ X → ℙ X → ℙ X := λ (h1: ℙ X) (h2: ℙ X), λ (x: X) (Y: U), and (h1 x Y) (h2 x Y)
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