Theorems · Theorem · general topology
Filter.atTop_neBot_iff
∀ {α : Type u_6} [inst : Preorder α], Filter.atTop.NeBot ↔ Nonempty α ∧ IsDirectedOrder α- Defined in
- Mathlib.Order.Filter.AtTopBot.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Filter.atTopstatement and proof · cited by 2,405
- Filter.NeBotstatement and proof · cited by 853
- IsDirectedOrderstatement and proof · cited by 316
- Filter.Eventually.existsproof · cited by 168
- Filter.Eventually.andproof · cited by 157
- Filter.eventually_ge_atTopproof · cited by 111
- Filter.nonempty_of_neBotproof · cited by 6
Cited by9
Results whose statement or proof uses this declaration.
- tendsto_atTop_ciSupproof · cited by 13
- MeasureTheory.tendsto_measure_iInter_atTopproof · cited by 12
- MeasureTheory.tendsto_measure_iUnion_atTopproof · cited by 10
- MeasureTheory.tendsto_measure_iUnion_accumulateproof · cited by 4
- Filter.tendsto_finset_Iic_atTop_atTopproof · cited by 2
- MeasureTheory.tendsto_setIntegral_of_antitoneproof · cited by 1
- MeasureTheory.tendsto_setIntegral_of_monotone₀proof · cited by 1
- MeasureTheory.tendsto_measure_iInter_leproof · cited by 1
- Submodule.starProjection_tendsto_closure_iSupproof · cited by 1