Theorems · Theorem · order theory
DenselyOrdered.dense
∀ {α : Type u_5} {inst : LT α} [self : DenselyOrdered α] (a₁ a₂ : α), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂An order is dense if there is an element between any pair of distinct elements.
- Defined in
- Mathlib.Order.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 2 from the axioms · uses no axioms
- Assumes
- DenselyOrdered
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites1
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DenselyOrderedstatement and proof · cited by 471
Cited by8
Results whose statement or proof uses this declaration.
- exists_betweenproof · cited by 102
- openSegment_sameproof · cited by 3
- Valuation.Integers.not_denselyOrdered_of_isPrincipalIdealRingproof · cited by 1
- DenselyOrdered.dense'proof · cited by 1
- WithBot.denselyOrdered_iffproof · cited by 1
- denselyOrdered_iff_denselyOrdered_units_and_nontrivial_unitsproof · cited by 1
- StrictConvex.eq_of_openSegment_subset_frontierproof · cited by 0
- WithTop.denselyOrdered_iffproof · cited by 0