LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.
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Every member of the generated monotone class intersects every original algebra member inside the generated class

Statement

Let $A$ be an algebra on $X$ and $M = m_{X} (A)$ . For every $B \in A$ and every $E \in M$ , one has $E \cap B \in M$ .

Facts & Assumptions

Given: An algebra $A$ on $X$ , its generated monotone class $M$ , and a fixed $B \in A$ .

[L1]

An algebra is closed under finite intersections (Algebras of subsets).

[L2]

The family $M$ is the smallest monotone class containing $A$ (The generated monotone class exists and is minimal).

Proof

technique · direct

1.1

Let $C_{B} : = {E \in M : E \cap B \in M}$ . Intersection with $B$ commutes with increasing unions and decreasing intersections, so the two monotone closure axioms for $M$ show that $C_{B}$ is a monotone class.

L2algebra

2.1

If $E \in A$ , then $E \cap B \in A \subseteq M$ by [L1] and [L2]. Thus $A \subseteq C_{B}$ , and minimality gives $M \subseteq C_{B}$ .

step 1.1L1L2∎

Depends on

Used by

The monotone class generated by an algebra is closed under finite intersections Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 4 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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