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

[benjub]

Add two lemmas relating to closed bounded intervals in a preorder:

• subset_Icc_iff : s ⊆ Icc a b ↔ a ∈ lowerBounds s ∧ b ∈ upperBounds s in Mathlib/Order/Bounds/Basic.lean characterizes containment in a closed bounded interval. This is the version "with witnesses" of bddBelow_bddAbove_iff_subset_Icc, and we prove the latter as a corollary.
• OrdConnected.eq_Icc (s : Set α) (hs : OrdConnected s) (ha : IsLeast s a) (hb : IsGreatest s b) : s = Icc a b in Mathlib/Order/Interval/Set/OrdConnected.lean gives a sufficient condition to be a closed bounded interval. It uses the previous lemma in its proof.

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Motivation: I plan to use OrdConnected.eq_Icc to show that a set is closed when it is ordconnected and contains its infimum and supremum. This will allow to use the existing "continuous induction principle" IsClosed.mem_of_ge_of_forall_exists_gt to shorten the (compiling) proof in the draft PR #41552.

[github-actions]

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[github-actions]

PR summary aa37d8c795

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ OrdConnected.eq_Icc
+ subset_Icc_iff

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit aa37d8c).

• +2 new declarations
• −0 removed declarations

+Set.OrdConnected.eq_Icc
+subset_Icc_iff

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit aa37d8c795
Reference commit 4efb186f10

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).