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feat(Order): add subset_Icc_iff and Set.OrdConnected.eq_Icc #41622

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feat(Order): add subset_Icc_iff and Set.OrdConnected.eq_Icc #41622

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8 changes: 5 additions & 3 deletions Mathlib/Order/Bounds/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -596,9 +596,11 @@ end
theorem bddBelow_iff_subset_Ici : BddBelow s ↔ ∃ a, s ⊆ Ici a :=
Iff.rfl

theorem bddBelow_bddAbove_iff_subset_Icc : BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Icc a b := by
simp [Ici_inter_Iic.symm, subset_inter_iff, bddBelow_iff_subset_Ici,
bddAbove_iff_subset_Iic, exists_and_left, exists_and_right]
theorem subset_Icc_iff : s ⊆ Icc a b ↔ (a ∈ lowerBounds s ∧ b ∈ upperBounds s) :=
subset_inter_iff.trans (mem_lowerBounds_iff_subset_Ici.and mem_upperBounds_iff_subset_Iic)

theorem bddBelow_bddAbove_iff_subset_Icc : BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Icc a b :=
exists_and_exists_comm.trans <| exists₂_congr fun _ _ ↦ subset_Icc_iff.symm

/-!
#### Univ
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4 changes: 4 additions & 0 deletions Mathlib/Order/Interval/Set/OrdConnected.lean
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Expand Up @@ -64,6 +64,10 @@ protected theorem Icc_subset (s : Set α) [hs : OrdConnected s] {x y} (hx : x
Icc x y ⊆ s :=
hs.out hx hy

theorem OrdConnected.eq_Icc {a b : α} (s : Set α) (hs : OrdConnected s) (ha : IsLeast s a)
(hb : IsGreatest s b) : s = Icc a b :=
(subset_Icc_iff.mpr ⟨ha.2, hb.2⟩).antisymm (hs.out ha.1 hb.1)

end Preorder

end Set
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