Theorems · Theorem · order theory
IsAtom.le_iff
∀ {α : Type u_2} [inst : PartialOrder α] [inst_1 : OrderBot α] {a x : α}, IsAtom a → (x ≤ a ↔ x = ⊥ ∨ x = a)- Defined in
- Mathlib.Order.Atoms
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 13 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.
Cites6
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
- le_iff_lt_or_eqproof · cited by 26
- IsAtom.lt_iffproof · cited by 5
Cited by10
Results whose statement or proof uses this declaration.
- IsAtom.le_iff_eqproof · cited by 8
- IsCoatom.le_iffproof · cited by 4
- IsAtom.Iic_eqproof · cited by 2
- Associates.isAtom_iffproof · cited by 2
- exists_sSupIndep_disjoint_sSup_atomsproof · cited by 2
- GaloisInsertion.isAtom_iffproof · cited by 2
- IsAtom.disjoint_of_neproof · cited by 1
- CompleteAtomicBooleanAlgebra.eq_setOfPred_le_sSup_and_isAtomproof · cited by 1
- IsAtom.ne_iff_eq_botproof · cited by 1
- Filter.le_pure_iff'proof · cited by 0