Theorems · Theorem · order theory
SetLike.le_def
∀ {A : Type u_1} {B : Type u_2} [inst : SetLike A B] [inst_1 : LE A] [IsConcreteLE A B] {S T : A},
S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T- Defined in
- Mathlib.Data.SetLike.Basic
- Cited by
- 76 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses propext, Quot.sound
- Assumes
- SetLikeLEIsConcreteLE
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- SetLikestatement and proof · cited by 1,084
- IsConcreteLEstatement and proof · cited by 28
Cited by76
Results whose statement or proof uses this declaration.
- mem_of_le_of_memproof · cited by 9
- AddSubmonoid.mem_iSup_of_memproof · cited by 6
- Submodule.span_univproof · cited by 6
- EuclideanGeometry.Sphere.IsTangentAt.dist_sq_eq_of_memproof · cited by 6
- EuclideanGeometry.oangle_eq_of_dist_orthogonalProjection_eqproof · cited by 5
- Submodule.singleton_span_isCompactElementproof · cited by 4
- UniqueDiffWithinAt.prodproof · cited by 4
- Affine.Simplex.touchpoint_mem_affineSpan_simplexproof · cited by 4
- collinear_iff_of_memproof · cited by 4
- AddSubmonoid.mem_sup_leftproof · cited by 3
- Finsupp.supported_eq_span_singleproof · cited by 3
- iSupIndep_iff_finsetSum_eq_zero_imp_eq_zeroproof · cited by 3