Theorems · Theorem · logic and foundations
Set.nonempty_iff_ne_empty
∀ {α : Type u} {s : Set α}, s.Nonempty ↔ s ≠ ∅See also Set.not_nonempty_iff_eq_empty.
- Defined in
- Mathlib.Data.Set.Basic
- Cited by
- 96 results in Mathlib
- Foundations
- Depth 26 from the axioms, rests on 146 definitions · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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.not_nonempty_iff_eq_emptyproof · cited by 56
- Iff.not_rightproof · cited by 20
Cited by96
Results whose statement or proof uses this declaration.
- Set.eq_empty_or_nonemptyproof · cited by 248
- Set.Nonempty.ne_emptyproof · cited by 65
- MeasureTheory.nonempty_of_measure_ne_zeroproof · cited by 22
- Set.ncard_posproof · cited by 11
- Set.Infinite.nonemptyproof · cited by 6
- convexHull_nonempty_iffproof · cited by 6
- Set.isEmpty_coe_sortproof · cited by 6
- Set.nonempty_of_encard_ne_zeroproof · cited by 5
- Set.nonempty_of_ncard_ne_zeroproof · cited by 5
- Set.pi_subset_pi_iffproof · cited by 4
- MeasureTheory.Measure.count_ne_zero_iffproof · cited by 4
- Function.support_nonempty_iffproof · cited by 4