Theorems · Theorem · logic and foundations
Cardinal.le_range_of_union_finset_eq_univ
∀ {α : Type u_1} {β : Type u_2} [Infinite β] (f : α → Finset β),
⋃ a, ↑(f a) = Set.univ → Cardinal.mk β ≤ Cardinal.mk ↑(Set.range f)If an infinite type β can be expressed as a union of finite sets,
then the cardinality of the collection of those finite sets
must be at least the cardinality of β.
- Defined in
- Mathlib.SetTheory.Cardinal.Pigeonhole
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Infinite
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Finsetstatement and proof · cited by 13,712
- SetLike.coestatement and proof · cited by 8,199
- Set.Elemstatement and proof · cited by 7,166
- Set.preimageproof · cited by 4,946
- Set.rangestatement and proof · cited by 4,705
- Set.univstatement and proof · cited by 3,945
- Cardinalstatement · cited by 2,598
- Set.iUnionstatement and proof · cited by 2,483
- Cardinal.mkstatement and proof · cited by 942
- Set.mem_univproof · cited by 416
- Eq.geproof · cited by 375
Cited by2
Results whose statement or proof uses this declaration.
- infinite_basis_le_maximal_linearIndependent'proof · cited by 2
- Cardinal.le_range_of_union_finset_eq_topproof · cited by 0