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

How statement and proof provenance work

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.

Recovering a prescribed starting point in DC

Statement

In ZF, the following implies DC with prescribed initial point: every serial relation on a nonempty set has an omega path, without specification of its first term. Thus the two versions of DC are equivalent.

Facts & Assumptions

[F1]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: The library version requires $x_{0} = a$ for every $a \in X$ .

Proof

Given: The objects and hypotheses in the statement.

1.1

Assume the version without a starting point. Fix a serial $R$ on $X \neq \emptyset$ and $a \in X$ . Let $T$ consist of nonempty finite $R$ -paths starting at $a$ . It contains $(a)$ . The relation of extending by exactly one term is serial, since the last point has a successor. Obtain a path $(t_{n})$ in $T$ under this relation.

given

2.1

The $t_{n}$ are nested and have lengths $len (t_{0}) + n$ . Their union has domain omega, starts at $a$ , and satisfies $x_{n} R x_{n + 1}$ at each index, since some finite path contains both coordinates. This is the prescribed-start version.

F1step 1.1

3.1

Conversely, on a nonempty $X$ fix one $a \in X$ and apply prescribed-start DC, then forget the value of its first term. This one existential selection does not invoke AC.

F1∎

Depends on

Used by

Dependent choices as extending partial tuples Example
DC and finite multiple selections Theorem

Dependency tree · two levels

11 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

Jech, The Axiom of Choice, §2.4, pp.22–23 (standard reference, not scraped)
alg-d, On dependent choice, Proposition 3, PDF p.2; Proposition 7, pp.4–5 (standard reference, not scraped)