Theorems · Theorem · logic and foundations
Set.Nonempty.ne_empty
∀ {α : Type u} {s : Set α}, s.Nonempty → s ≠ ∅Alias of the forward direction of Set.nonempty_iff_ne_empty.
See also Set.not_nonempty_iff_eq_empty.
- Defined in
- Mathlib.Data.Set.Basic
- Cited by
- 65 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
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 · cited by 2,627
- Set.nonempty_iff_ne_emptyproof · cited by 96
Cited by65
Results whose statement or proof uses this declaration.
- Set.prod_subset_prod_iffproof · cited by 15
- Set.exists_eq_singleton_iff_nonempty_subsingletonproof · cited by 9
- Set.singleton_ne_emptyproof · cited by 8
- Nat.sInf_eq_zeroproof · cited by 6
- Filter.compl_notMemproof · cited by 6
- isPreconnected_closed_iffproof · cited by 6
- IsClopen.eq_univproof · cited by 5
- Filter.empty_notMemproof · cited by 5
- Set.Nonempty.subset_singleton_iffproof · cited by 5
- UpperSet.Ici_ne_topproof · cited by 4
- Filter.not_isBoundedUnder_of_tendsto_atTopproof · cited by 4
- Vitali.exists_disjoint_subfamily_covering_enlargementproof · cited by 3