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Theorems · Theorem · order theory

le_iff_transGen_wcovBy

∀ {α : Type u_2} [inst : Preorder α] [LocallyFiniteOrder α] {x y : α},
  x ≤ y ↔ Relation.TransGen (fun x1 x2 => x1 ⩿ x2) x y

In a locally finite preorder, ≤ is the transitive closure of ⩿.

Defined in
Mathlib.Order.Interval.Finset.Basic
Cited by
2 results in Mathlib
Foundations
Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
PreorderLocallyFiniteOrder
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

le_iff_reflTransGen_covBy · cited by 1le_iff_reflTransGen_covBymonotone_iff_forall_wcovBy · cited by 1monotone_iff_forall_wcovByPreorder · cited by 7952PreorderLE.le.trans · cited by 3151le.transLocallyFiniteOrder · cited by 658LocallyFiniteOrderWCovBy · cited by 107WCovByWCovBy.le · cited by 15WCovBy.letransGen_wcovBy_of_le · cited by 1transGen_wcovBy_of_lele_iff_transGen_wcovByCITED BYCITES

Cites6

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Cited by2

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