TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The diagonal intersection of clubs is club

Statement

In ZFC, if $κ$ is regular uncountable and $(C_{ξ})_{ξ < κ}$ is a sequence of clubs of $κ$ , then $D = △_{ξ < κ} C_{ξ}$ is club.

Facts & Assumptions

[F1]

Diagonal intersection and union: Membership at alpha tests only indices xi<alpha.

[F2]

Closure points form a club: Any self-map of a regular uncountable cardinal has club many closure points.

Proof

Given: The objects and hypotheses in the statement.

1.1

Define $g (β) = \sup_{ξ \leq β} \min (C_{ξ} ∖ (β + 1))$ . Regularity keeps this below $κ$ . By the closure-point lemma, there are unboundedly many nonzero closure points $α$ of $g$ ; these are limits since $g (β) > β$ . Fix $ξ < α$ and any $η < α$ , then take $β < α$ with $β \geq ξ, η$ . The least $C_{ξ}$ point above $β$ is below $α$ . Thus $α \in C_{ξ}$ , proving $α \in D$ .

F1F2

2.1

If $δ$ is a nonzero limit point of $D$ , then for each $ξ < δ$ the points of $D \cap δ$ above $ξ$ belong to $C_{ξ}$ and are unbounded in $δ$ . Its closure gives $δ \in C_{ξ}$ for every $ξ < δ$ , hence $δ \in D$ . This proves closure. Zero is in $D$ by definition.

F1step 1.1∎

Depends on

Used by

The club filter is the least normal tail filter Corollary
Tails, limits, and diagonal intersection Example
Splitting a stationary set concentrated on regular cardinals Lemma
Fodor’s pressing-down lemma Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lietz, Theorem 5.9, p.41 (standard reference, not scraped)
Vasey, Theorem 14.11, p.81 (standard reference, not scraped)