Hausdorff's maximality principle
ProvedFamousTheorems.maxchain_specorder-theoryset-theoryzorn
Hausdorff's maximality principle. Every partially ordered set contains a maximal chain -- a totally ordered subset not properly contained in another. It is equivalent to Zorn's lemma and to the axiom of choice, and it is often the more convenient entry point: one obtains a maximal chain first and then takes its supremum, which is exactly how Zorn's lemma is usually derived from it. Hausdorff published it in 1914, well before Zorn's 1935 paper. Formalization note. maxChain is the constructed maximal chain and the statement asserts its defining property. The result is Mathlib's maxChain_spec.
Preamble
import Mathlib
Formal statement
namespace FamousTheorems
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25
open Filter Set Topology DirectSum
theorem maxchain_spec :
∀ {α : Type u_1} {r : α → α → Prop}, IsMaxChain r (maxChain r) := by sorry
end FamousTheoremsSource
Marked as a named theorem in Mathlib's own docstrings; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.