Theorems · Theorem · order theory
Set.nonempty_Ioc
∀ {α : Type u_1} [inst : Preorder α] {a b : α}, (Set.Ioc b a).Nonempty ↔ b < a- Defined in
- Mathlib.Order.Interval.Set.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
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.
- Preorderstatement and proof · cited by 7,952
- Set.Nonemptystatement and proof · cited by 2,627
- Set.Iocstatement and proof · cited by 971
- LE.le.trans_lt'proof · cited by 50
- Set.right_mem_Iocproof · cited by 18
Cited by9
Results whose statement or proof uses this declaration.
- Metric.cthickening_eq_iInter_thickeningproof · cited by 3
- BoxIntegral.Prepartition.injOn_setOfPred_mem_Icc_setOfPred_lower_eqproof · cited by 2
- Finset.nonempty_Iocproof · cited by 1
- isConnected_Iocproof · cited by 1
- Set.Ioc_eq_empty_iffproof · cited by 1
- Metric.cthickening_eq_iInter_cthickeningproof · cited by 1
- left_nhdsWithin_Ioc_neBotproof · cited by 0
- Set.nonempty_Ioc_subtypeproof · cited by 0
- csInf_Iocproof · cited by 0