LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

AC supplies the dependent-choice instances used in vector-bundle constructions

Statement

In ZF, the Axiom of Choice implies the prescribed-initial-point form of the Axiom of Dependent Choice. Explicitly, if $X$ is nonempty, $R$ is an entire relation on $X$ , and $a \in X$ , then AC supplies a sequence $x : N \to X$ such that

$x_{0} = a and x_{n} R x_{n + 1}$

for every $n \in N$ . Consequently, a vector-bundle theorem stated under AC may discharge a separately declared DC hypothesis without assuming another choice principle.

Facts & Assumptions

Given: ZF, AC, a nonempty set $X$ , a relation $R$ entire on $X$ , and a prescribed point $a \in X$ .

[F1]

AC says that every family of nonempty sets has a choice function (The Axiom of Choice).

[F2]

DC with prescribed initial point asks for a function $x : N \to X$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ for every $n$ (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F3]

Given a set $X$ , a point $a \in X$ , and a function $s : X \to X$ , recursion on $N$ supplies a unique function $x : N \to X$ with $x_{0} = a$ and $x_{n + 1} = s (x_{n})$ (The recursion theorem).

Proof

technique · direct

1.1F1givenconstruct

For each $u \in X$ , let $S_{u} = {v \in X : u R v}$ . Every $S_{u}$ is nonempty because $R$ is entire. Apply [F1] to the set $S = {S_{u} : u \in X}$ and let $c$ be its choice function. Define $s : X \to X$ by $s (u) = c (S_{u})$ . This is well-defined even when two successor sets coincide, and $u R s (u)$ for every $u \in X$ .

2.1F3step 1.1

Apply [F3] to $s$ and the prescribed $a$ . It gives $x : N \to X$ with $x_{0} = a$ and $x_{n + 1} = s (x_{n})$ . Step 1.1 then gives $x_{n} R x_{n + 1}$ for every $n$ .

3.1F2step 1.1step 2.1∎

Since $X$ , the entire relation $R$ , and $a$ were arbitrary, the sequence in step 2.1 satisfies exactly the prescribed-initial-point formulation in [F2]. Thus AC implies DC, and every later use of this lemma spends AC only in the simultaneous choice made in step 1.1.

Depends on

Dependency tree · 2 levels

Sources

Thomas J. Jech, The Axiom of Choice, §2.4.1