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

Dependency tree · 2 levels

Sources

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.9 and Theorem 2.10