Theorems · Theorem · general topology
Filter.nonempty_of_mem
∀ {α : Type u} {f : Filter α} [hf : f.NeBot] {s : Set α}, s ∈ f → s.Nonempty- Defined in
- Mathlib.Order.Filter.Basic
- Cited by
- 36 results in Mathlib
- Foundations
- Depth 52 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Filter.NeBot
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Filterstatement and proof · cited by 8,121
- Set.Nonemptystatement · cited by 2,627
- Filter.NeBotstatement and proof · cited by 853
- Set.eq_empty_or_nonemptyproof · cited by 248
- Filter.empty_mem_iff_botproof · cited by 18
- Filter.NeBot.ne'proof · cited by 7
Cited by36
Results whose statement or proof uses this declaration.
- Filter.exists_seq_tendstoproof · cited by 21
- Filter.NeBot.nonempty_of_memproof · cited by 18
- Filter.nonempty_of_neBotproof · cited by 6
- Filter.compl_notMemproof · cited by 6
- Asymptotics.IsLittleOTVS.zeroproof · cited by 5
- Filter.empty_notMemproof · cited by 5
- Filter.not_isBoundedUnder_of_tendsto_atTopproof · cited by 4
- Filter.forall_mem_nonempty_iff_neBotproof · cited by 4
- balancedCore_mem_nhds_zeroproof · cited by 4
- image_le_of_liminf_slope_right_lt_deriv_boundary'proof · cited by 4
- Asymptotics.IsLittleOTVS.smul_leftproof · cited by 3
- Ultrafilter.nonempty_of_memproof · cited by 3