Theorems · Theorem · logic and foundations
Cardinal.infinite_pigeonhole
∀ {β α : Type u} (f : β → α),
Cardinal.aleph0 ≤ Cardinal.mk β →
Cardinal.mk α < (Cardinal.mk β).ord.cof → ∃ a, Cardinal.mk ↑(f ⁻¹' {a}) = Cardinal.mk βThe infinite pigeonhole principle
- Defined in
- Mathlib.SetTheory.Cardinal.Pigeonhole
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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.Elemstatement and proof · cited by 7,166
- Set.preimagestatement and proof · cited by 4,946
- Cardinalstatement · cited by 2,598
- Cardinal.mkstatement and proof · cited by 942
- LE.le.trans_ltproof · cited by 795
- LT.lt.trans_leproof · cited by 678
- Cardinal.aleph0statement and proof · cited by 521
- Cardinal.ordstatement and proof · cited by 266
- Ordinal.cofstatement and proof · cited by 125
- LE.le.antisymm'proof · cited by 104
- Set.preimage_univproof · cited by 46
Cited by1
Results whose statement or proof uses this declaration.
- Cardinal.infinite_pigeonhole_cardproof · cited by 4