[Merged by Bors] - feat(logic/function/basic): not_surjective_Type #12311
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ChrisHughes24
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| @@ -213,6 +213,23 @@ theorem cantor_injective {α : Type*} (f : (set α) → α) : | |||
| cantor_surjective (λ a b, ∀ U, a = f U → U b) $ | |||
| right_inverse.surjective (λ U, funext $ λ a, propext ⟨λ h, h U rfl, λ h' U' e, i e ▸ h'⟩) | |||
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| /-- There is no surjection from `α : Type u` into `Type u`. This theorem | |||
| demonstrates why `Type : Type` would be inconsistent in Lean. -/ | |||
| theorem not_surjective_Type {α : Type u} (f : α → Type (max u v)) : | |||
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If you are willing to move the theorem to logic/small.lean, then the proof is just
not_small_type ∘ small_of_surjective
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That file has a lot of dependencies, comparatively speaking. Perhaps the proof can be used to shorten this one, but I think it is fine where it is.
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Zulip discussion. @digama0 suggested this theorem should be in mathlib. The proof is slightly more complicated than it is in the discussion because certain simp lemmas aren't available at this point in mathlib.