LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

The generated monotone class exists and is minimal

Statement

For every set $X$ and every $E \subseteq P (X)$ , the family $m_{X} (E)$ is a monotone class on $X$ , contains $E$ , and is contained in every monotone class on $X$ that contains $E$ .

Facts & Assumptions

Given: A set $X$ , a family $E \subseteq P (X)$ , and the intersection definition of $m_{X} (E)$ in The monotone class generated by a family of sets.

Proof

technique · direct

1.1given

A nonempty intersection of monotone classes is closed under increasing countable unions and decreasing countable intersections, because each operation is performed in every class being intersected.

1.2givenconstruct

The power set $P (X)$ is a monotone class containing $E$ , so the family intersected in the definition of $m_{X} (E)$ is nonempty.

2.1step 1.1step 1.2∎

By steps 1.1 and 1.2, $m_{X} (E)$ is a monotone class. Every generator belongs to every class in the intersection, and the intersection is contained in each such class; hence it contains $E$ and is minimal.

Depends on

The monotone class generated by a family of sets

Used by

Cited to discharge well-definedness by The monotone class generated by a family of sets.

Dependency tree · two levels

2 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.9 and Theorem 2.10 (standard reference, not scraped)