Theorems · Theorem · order theory
exists_lt_of_lt_csSup
∀ {α : Type u_1} [inst : ConditionallyCompleteLinearOrder α] {s : Set α} {b : α},
s.Nonempty → b < sSup s → ∃ a ∈ s, b < aWhen b < sSup s, there is an element a in s with b < a, if s is nonempty and the order
is a linear order.
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Classical.choice, Quot.sound
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 and proof · cited by 53,352
- Set.Nonemptystatement and proof · cited by 2,627
- SupSet.sSupstatement and proof · cited by 954
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- csSup_leproof · cited by 35
Cited by13
Results whose statement or proof uses this declaration.
- Filter.eventually_lt_of_lt_liminfproof · cited by 17
- Antitone.map_limsSup_of_continuousAtproof · cited by 6
- exists_lt_of_lt_ciSupproof · cited by 4
- csSup_mem_of_not_isSuccLimitproof · cited by 3
- MonotoneOn.tendsto_nhdsLTproof · cited by 3
- Vitali.exists_disjoint_subfamily_covering_enlargementproof · cited by 3
- MeasureTheory.hahn_decompositionproof · cited by 2
- MonotoneOn.tendsto_nhdsWithin_Ioo_leftproof · cited by 2
- Real.add_neg_lt_sSupproof · cited by 1
- Besicovitch.TauPackage.mem_iUnionUpTo_lastStepproof · cited by 1
- Vitali.exists_disjoint_covering_aeproof · cited by 1
- MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurableproof · cited by 1