Theorems · Theorem · order theory
IsChain.directedOn
∀ {α : Type u_1} {r : α → α → Prop} {s : Set α} [Std.Refl r], IsChain r s → DirectedOn r s- Defined in
- Mathlib.Order.Preorder.Chain
- Cited by
- 28 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Std.Refl
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- DirectedOnstatement · cited by 271
- IsChainstatement and proof · cited by 158
- reflproof · cited by 80
- IsChain.totalproof · cited by 12
Cited by28
Results whose statement or proof uses this declaration.
- Ideal.radical_eq_sInfproof · cited by 21
- exists_linearIndepOn_extensionproof · cited by 5
- Vitali.exists_disjoint_subfamily_covering_enlargementproof · cited by 3
- exists_idempotent_of_compact_t2_of_continuous_mul_leftproof · cited by 2
- exists_sSupIndep_disjoint_sSup_atomsproof · cited by 2
- exists_maximal_orthonormalproof · cited by 2
- exists_idempotent_of_compact_t2_of_continuous_add_leftproof · cited by 2
- RieszExtension.exists_topproof · cited by 1
- IsChain.pairwise_iUnion₂proof · cited by 1
- IsChain.pairwise_sUnionproof · cited by 1
- IsClosed.exists_minimal_nonempty_closed_subsetproof · cited by 1
- Topology.IsScott.isOpen_iff_Iic_compl_or_univproof · cited by 1