Theorems · Theorem · order theory
isLUB_le_iff
∀ {α : Type u_1} [inst : Preorder α] {s : Set α} {a b : α}, IsLUB s a → (a ≤ b ↔ b ∈ upperBounds s)- Defined in
- Mathlib.Order.Bounds.Basic
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
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
- Preorderstatement and proof · cited by 7,952
- IsLUBstatement and proof · cited by 280
- upperBoundsstatement · cited by 263
- IsLUB.upperBounds_eqproof · cited by 10
Cited by24
Results whose statement or proof uses this declaration.
- iSup_le_iffproof · cited by 45
- sSup_le_iffproof · cited by 8
- lt_isLUB_iffproof · cited by 7
- Order.IsNormal.le_iff_forall_leproof · cited by 6
- ciSup_le_iffproof · cited by 6
- csSup_le_iff'proof · cited by 5
- le_of_isLUB_Iioproof · cited by 3
- norm_cfc_leproof · cited by 3
- norm_cfcₙ_leproof · cited by 3
- csSup_le_iffproof · cited by 2
- Directed.ciSup_le_iffproof · cited by 2
- Sion.minimaxproof · cited by 2