Theorems · Theorem · order theory
isLUB_iff_le_iff
∀ {α : Type u_1} [inst : Preorder α] {s : Set α} {a : α}, IsLUB s a ↔ ∀ (b : α), a ≤ b ↔ b ∈ upperBounds s- Defined in
- Mathlib.Order.Bounds.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- le_rflproof · cited by 1,558
- IsLUBstatement and proof · cited by 280
- upperBoundsstatement and proof · cited by 263
- isLUB_le_iffproof · cited by 24
Cited by1
Results whose statement or proof uses this declaration.
- MonotoneOn.sSup_image_Iccproof · cited by 1