Existing in Mathlib · Combinatorics

Sperner’s Theorem

Take a finite collection of subsets in which no selected set contains another selected set. Sperner’s theorem says that the collection cannot be larger than the family of all subsets having the middle size.

antichains
Boolean lattice
middle layer
set-family shadows
LYM inequality

Exact theorem

Exact Mathlib statement

theorem IsAntichain.sperner {α : Type*} [Fintype α] {𝒜 : Finset (Finset α)} (h𝒜 : IsAntichain (fun s t => s ⊆ t) (SetLike.coe 𝒜)) : 𝒜.card ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)

The theorem at a glance

Sperner’s Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An editorial theorem poster relates antichains, Boolean-lattice rank layers, shadows, the LYM inequality, and the middle-layer upper bound. — Shadows and the LYM density bound force every finite antichain below the size of the Boolean lattice’s middle layer.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A Boolean lattice widens through rank layers to a highlighted middle layer whose nodes are pairwise incomparable. — No antichain can contain more subsets than a middle rank of the Boolean lattice.

Why it matters

A mathematical landmark

Sperner’s theorem is a foundational extremal result about the Boolean lattice. Mathlib’s proof reaches it through shadows and the Lubell–Yamamoto–Meshalkin inequality, giving the page a strong source-anchored proof story.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib’s finite-subset-poset form of Sperner’s theorem. The ground type is finite, the family is a Finset of Finsets, and incomparability is under subset inclusion. The selected declaration proves only the sharp numerical upper bound; it does not classify equality cases or say that every antichain is a rank layer.

ProofAtlas did not originate Sperner’s theorem or Mathlib’s declaration.
The selected declaration is not a theorem about arbitrary infinite set families.
The selected declaration does not classify the equality cases.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: IsAntichain.sperner in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.sperner-theorem.v001
Source: Open the pinned upstream reference