Theorems · Theorem · order theory
IsCompactlyGenerated.BooleanGenerators.mem_of_isAtom_of_le_sSup_atoms
∀ {α : Type u_1} [inst : CompleteLattice α] {S : Set α} [IsCompactlyGenerated α],
IsCompactlyGenerated.BooleanGenerators S → ∀ (a : α), IsAtom a → a ≤ sSup S → a ∈ S- Defined in
- Mathlib.Order.BooleanGenerators
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Set.Nonemptyproof · cited by 2,627
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement and proof · cited by 954
- Set.eq_empty_or_nonemptyproof · cited by 248
- IsAtomstatement and proof · cited by 130
- le_sSupproof · cited by 79
- IsCompactlyGeneratedstatement and proof · cited by 37
- sSup_emptyproof · cited by 19
- IsCompactlyGenerated.BooleanGeneratorsstatement and proof · cited by 11
- IsAtom.le_iff_eqproof · cited by 8
- IsCompactlyGenerated.BooleanGenerators.isAtomproof · cited by 6
Cited by3
Results whose statement or proof uses this declaration.
- IsCompactlyGenerated.BooleanGenerators.sSup_le_sSup_iff_of_atomsproof · cited by 0
- IsCompactlyGenerated.BooleanGenerators.eq_atoms_of_sSup_eq_topproof · cited by 0
- IsCompactlyGenerated.BooleanGenerators.sSup_interproof · cited by 0