bookshelf/Common/Set/Intervals.lean

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import Common.Logic.Basic
import Mathlib.Data.Set.Intervals.Basic
namespace Set
/-! # Common.Set.Intervals
Additional theorems around intervals.
-/
theorem Iio_nat_lt_ssubset {m n : } (h : m < n)
: Iio m ⊂ Iio n := by
rw [ssubset_def]
apply And.intro
· unfold Iio
simp only [setOf_subset_setOf]
intro x hx
calc x
_ < m := hx
_ < n := h
· show ¬ ∀ x, x < n → x < m
simp only [not_forall, not_lt, exists_prop]
exact ⟨m, h, by simp⟩
end Set