Theorems · Theorem · combinatorics
Finite.exists_ne_map_eq_of_infinite
∀ {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] (f : α → β), ∃ x y, x ≠ y ∧ f x = f yThe pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are
infinitely many pigeons in finitely many pigeonholes, then there are at least two pigeons in the
same pigeonhole.
See also: Fintype.exists_ne_map_eq_of_card_lt, Finite.exists_infinite_fiber.
- Defined in
- Mathlib.Data.Fintype.Pigeonhole
- Cited by
- 5 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.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finitestatement and proof · cited by 3,029
- Infinitestatement and proof · cited by 352
- not_injective_infinite_finiteproof · cited by 7
Cited by5
Results whose statement or proof uses this declaration.
- OreLocalization.oreDiv_one_surjective_of_finite_leftproof · cited by 2
- OreLocalization.oreDiv_one_surjective_of_finite_rightproof · cited by 2
- AddOreLocalization.oreSub_zero_surjective_of_finite_leftproof · cited by 2
- AddOreLocalization.oreSub_zero_surjective_of_finite_rightproof · cited by 2
- Field.isAlgebraic_of_finite_intermediateFieldproof · cited by 2