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Theorems · Theorem · order theory

Set.nonempty_Ico

∀ {α : Type u_1} [inst : Preorder α] {a b : α}, (Set.Ico a b).Nonempty ↔ a < b
Defined in
Mathlib.Order.Interval.Set.Basic
Cited by
6 results in Mathlib
Foundations
Depth 8 from the axioms · uses propext
Assumes
Preorder
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Finset.nonempty_Ico · cited by 1Finset.nonempty_IcoSet.Ico_eq_empty_iff · cited by 1Set.Ico_eq_empty_iffSet.nonempty_Ico_subtype · cited by 0Set.nonempty_Ico_subtypecsSup_Ico · cited by 0csSup_IcoisConnected_Ico · cited by 0isConnected_Icoright_nhdsWithin_Ico_neBot · cited by 0right_nhdsWithin_Ico_neBotPreorder · cited by 7952PreorderSet.Nonempty · cited by 2627Set.NonemptySet.Ico · cited by 799Set.IcoLE.le.trans_lt · cited by 795le.trans_ltSet.left_mem_Ico · cited by 18Set.left_mem_IcoSet.nonempty_IcoCITED BYCITES

Cites5

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by6

Results whose statement or proof uses this declaration.