Theorems · Theorem · logic and foundations
Cardinal.exists_infinite_fiber
∀ {β α : Type u} (f : β → α), Cardinal.mk α < Cardinal.mk β → ∀ [Infinite β], ∃ a, Infinite ↑(f ⁻¹' {a})A function whose domain's cardinality is infinite and strictly greater than its codomain's has an infinite fiber.
- Defined in
- Mathlib.SetTheory.Cardinal.Pigeonhole
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 106 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.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Set.Elemstatement and proof · cited by 7,166
- Set.preimagestatement and proof · cited by 4,946
- LE.le.transproof · cited by 3,151
- Cardinalstatement and proof · cited by 2,598
- LT.lt.leproof · cited by 2,189
- le_rflproof · cited by 1,558
- Cardinal.mkstatement and proof · cited by 942
- Cardinal.aleph0proof · cited by 521
- Infinitestatement and proof · cited by 352
- lt_or_geproof · cited by 182
- Cardinal.aleph0_le_mkproof · cited by 32
Cited by2
Results whose statement or proof uses this declaration.
- Cardinal.le_range_of_union_finset_eq_univproof · cited by 2
- Cardinal.exists_infinite_fiber'proof · cited by 1