TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Member equivalence is transitive

Statement

For members of one object in an abelian category, the relation $\equiv$ is transitive.

Facts & Assumptions

Given: An abelian category and members $x : X \to A$ , $y : Y \to A$ , and $z : Z \to A$ with $x \equiv y$ and $y \equiv z$ .

[L1]

The relation $x \equiv y$ is witnessed by epimorphisms from one common domain as in Equivalence of members.

[L2]

Pullbacks exist in an abelian category (Pullbacks and pushouts as limits and colimits of cospans and spans).

[L3]

The pullback of an epimorphism is an epimorphism (The pullback of an epimorphism is an epimorphism).

Proof

technique · direct

1.1

Choose epimorphisms $u : W_{1} \to X$ , $v : W_{1} \to Y$ , $w : W_{2} \to Y$ , and $r : W_{2} \to Z$ with $x u = y v$ and $y w = z r$ , using [L1].

L1choose

1.2

Form the pullback of $v$ and $w$ , with projections $v^{'} : P \to W_{1}$ and $w^{'} : P \to W_{2}$ . By [L3], both $v^{'}$ and $w^{'}$ are epic.

L2L3construct

2.1

The pullback equation gives $v v^{'} = w w^{'}$ , so $x u v^{'} = y v v^{'} = y w w^{'} = z r w^{'}$ . Hence the epimorphisms $u v^{'} : P \to X$ and $r w^{'} : P \to Z$ witness $x \equiv z$ .

L1step 1.1step 1.2algebra

3.1

Therefore member equivalence is transitive.

step 2.1∎

Depends on

Used by

FALSE: member equivalence is transitive in any pointed category with pullbacks False statement
Each object has a zero member and each member has a negative Proposition
The cost of the member calculus Remark
Members modulo equivalence correspond to subobjects Theorem
The subtraction surrogate 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

Saunders Mac Lane, Categories for the Working Mathematician, Proposition 2 and Theorem 3 (standard reference, not scraped)