3.5 subsets
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@ -329,9 +329,40 @@ module definition3∙4∙3 where
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## 3.5 Subsets and propositional resizing
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## 3.5 Subsets and propositional resizing
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### Lemma 3.5.1
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```
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lemma3∙5∙1 : {A : Set} {P : A → Set}
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→ ((x : A) → isProp (P x))
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→ (u v : Σ A P)
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→ (Σ.fst u ≡ Σ.fst v)
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→ u ≡ v
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lemma3∙5∙1 {P = P} f u v p =
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let
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eqv = theorem2∙7∙2 u v
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func = Σ.fst (lemma2∙4∙12.sym-equiv eqv)
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prf = p , f (Σ.fst v) (transport P p (Σ.snd u)) (Σ.snd v)
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in func prf
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```
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```
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```
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```
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SubProp : (l : Level) → Set (lsuc l)
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SubProp l = Σ (Set l) isProp
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```
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```
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equation3∙5∙4 : {l : Level} → SubProp l → SubProp (lsuc l)
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equation3∙5∙4 {l} (A , Aprop) = Lift A , λ x y → ap lift (Aprop (Lift.lower x) (Lift.lower y))
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```
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### Axiom 3.5.5
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Not able to be proved, but is consistent
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```
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postulate
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propResizingEquiv : {l : Level} → isequiv (equation3∙5∙4 {l})
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```
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## 3.6 The logic of mere propositions
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## 3.6 The logic of mere propositions
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