Theorems · Theorem · order theory
IsAtom.le_iff_eq
∀ {α : Type u_2} [inst : PartialOrder α] [inst_1 : OrderBot α] {a b : α}, IsAtom a → b ≠ ⊥ → (b ≤ a ↔ b = a)- Defined in
- Mathlib.Order.Atoms
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 14 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PartialOrderOrderBot
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.
- PartialOrderstatement and proof · cited by 6,410
- Bot.botstatement and proof · cited by 4,720
- OrderBotstatement and proof · cited by 1,055
- IsAtomstatement and proof · cited by 130
- IsAtom.le_iffproof · cited by 10
Cited by8
Results whose statement or proof uses this declaration.
- IsCoatom.le_iff_eqproof · cited by 4
- IsCompactlyGenerated.BooleanGenerators.mem_of_isAtom_of_le_sSup_atomsproof · cited by 3
- RootedTree.mem_subtrees_disjoint_iffproof · cited by 1
- lie_eq_self_of_isAtom_of_ne_botproof · cited by 1
- Submodule.isAtom_iff_finrank_eq_oneproof · cited by 1
- Submodule.linearEquiv_of_le_sSupproof · cited by 1
- Submodule.linearEquiv_of_sSup_eq_topproof · cited by 1
- TopologicalSpace.Closeds.coe_eq_singleton_of_isAtomproof · cited by 0