Theorems · Theorem · general topology
Filter.empty_mem_iff_bot
∀ {α : Type u} {f : Filter α}, ∅ ∈ f ↔ f = ⊥- Defined in
- Mathlib.Order.Filter.Basic
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 51 from the axioms · uses propext, Quot.sound
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
- Bot.botstatement and proof · cited by 4,720
- Filter.mem_of_supersetproof · cited by 308
- Set.empty_subsetproof · cited by 70
- bot_uniqueproof · cited by 57
- Filter.mem_botproof · cited by 7
Cited by18
Results whose statement or proof uses this declaration.
- Filter.nonempty_of_memproof · cited by 36
- isCompact_emptyproof · cited by 17
- Filter.map_eq_bot_iffproof · cited by 8
- Filter.filter_eq_bot_of_isEmptyproof · cited by 7
- CovBy.nhdsGTproof · cited by 4
- MeasureTheory.ae_eq_botproof · cited by 4
- isLindelof_emptyproof · cited by 3
- Filter.inf_principal_eq_botproof · cited by 3
- CovBy.nhdsLTproof · cited by 2
- Bornology.cobounded_eq_bot_iffproof · cited by 2
- Filter.eventually_false_iff_eq_botproof · cited by 2
- Filter.principal_eq_bot_iffproof · cited by 1