Theorems · Theorem · order theory
Nat.eq_Ici_of_nonempty_of_upward_closed
∀ {s : Set ℕ}, s.Nonempty → (∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) → s = Set.Ici (sInf s)- Defined in
- Mathlib.Order.Lattice.Nat
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Classical.choice, 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
- Set.Nonemptystatement and proof · cited by 2,627
- Set.extproof · cited by 2,266
- Set.Icistatement and proof · cited by 1,070
- InfSet.sInfstatement and proof · cited by 935
- Nat.sInf_leproof · cited by 20
- Nat.sInf_memproof · cited by 15
Cited by1
Results whose statement or proof uses this declaration.
- Nat.sInf_upward_closed_eq_succ_iffproof · cited by 3