ordConnected_iff_of_bdd — Mathlib

English

dual_ordSeparatingSet: ordSeparatingSet (ofDual^{-1} s) (ofDual^{-1} t) = ofDual^{-1} (ordSeparatingSet s t).

Русский

двойственная сепарационная установка: ordSeparatingSet(ор равной) ...

LaTeX

$$dual_ordSeparatingSet : ordSeparatingSet (ofDual^{-1} \,' s) (ofDual^{-1} \,' t) = ofDual^{-1} (ordSeparatingSet s t)$$

Lean4

theorem ordConnected_iff_of_bdd [ConditionallyCompleteLinearOrder α] [LocallyFiniteOrder α] (h₀ : I.Nonempty)
    (h₁ : BddBelow I) (h₂ : BddAbove I) : I.OrdConnected ↔ I = Icc (sInf I) (sSup I) :=
  have h₄ : I.Finite := h₁.finite_of_bddAbove h₂
  ⟨fun _ ↦ le_antisymm (subset_Icc_csInf_csSup h₁ h₂) (I.Icc_subset (h₀.csInf_mem h₄) (h₀.csSup_mem h₄)), fun h₃ ↦
    h₃ ▸ ordConnected_Icc⟩

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