Theorems · Theorem · order theory
Set.nonempty_Ioo
∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty ↔ a < b- Defined in
- Mathlib.Order.Interval.Set.Basic
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
- Assumes
- PreorderDenselyOrdered
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Preorderstatement and proof · cited by 7,952
- Set.Nonemptystatement and proof · cited by 2,627
- Set.Ioostatement and proof · cited by 1,214
- DenselyOrderedstatement and proof · cited by 471
- LT.lt.transproof · cited by 370
- exists_betweenproof · cited by 102
Cited by23
Results whose statement or proof uses this declaration.
- closure_Iooproof · cited by 20
- exists_seq_strictMono_tendsto'proof · cited by 6
- Dense.exists_betweenproof · cited by 5
- intervalIntegral.integral_deriv_smul_comp'''proof · cited by 4
- csInf_Iooproof · cited by 4
- exists_Ioo_extr_on_Iccproof · cited by 3
- Real.surjOn_tanproof · cited by 2
- Set.Ioo_eq_empty_iffproof · cited by 2
- tendsto_of_no_upcrossingsproof · cited by 2
- Convex.nontrivial_iff_nonempty_interiorproof · cited by 2
- MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersetsproof · cited by 1
- Set.nonempty_Ioo_subtypeproof · cited by 1