Theorems · Theorem · order theory
Set.Ioo_eq_empty_iff
∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], Set.Ioo a b = ∅ ↔ ¬a < b- Defined in
- Mathlib.Order.Interval.Set.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PreorderDenselyOrdered
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.Ioostatement · cited by 1,214
- DenselyOrderedstatement and proof · cited by 471
- Set.nonempty_Iooproof · cited by 23
Cited by2
Results whose statement or proof uses this declaration.
- Finset.Ioo_eq_empty_iffproof · cited by 1
- MeasureTheory.Measure.measure_Ioo_eq_zeroproof · cited by 0