Theorems · Definition · order theory
IsChain.linearOrder
{α : Type u_1} →
[inst : PartialOrder α] → [DecidableLE α] → {s : Set α} → IsChain (fun x1 x2 => x1 ≤ x2) s → LinearOrder ↑sA chain in a partial order is a linear order.
- Defined in
- Mathlib.Order.Preorder.Chain
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PartialOrderDecidableLE
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
- LinearOrderstatement · cited by 8,572
- Set.Elemstatement and proof · cited by 7,166
- PartialOrderstatement and proof · cited by 6,410
- IsChainstatement and proof · cited by 158
- decidableEqOfDecidableLEproof · cited by 0
- decidableLTOfDecidableLEproof · cited by 0
Cited by1
Results whose statement or proof uses this declaration.
- Filter.EventuallyConst.of_monotone_of_lt_cofproof · cited by 3