DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Transitive closure of a set

Definition

For a set $a$ , form $a_{0} = a$ and $a_{n + 1} = a_{n} \cup ⋃ a_{n}$ by recursion on $ω$ . Set

$TC (a) = ⋃_{n \in ω} a_{n} .$

This is a set by the same recursion, Replacement and Union construction as finite predecessor closure, now starting from $a$ rather than a singleton and using membership as the setlike relation. The convention is the least transitive superset of $a$ ; to include $a$ itself as an element, use $TC ({a})$ . This construction does not assume Foundation.

Conventions and prerequisites: Finite predecessor closures are sets.

Depends on

Finite predecessor closures are sets

Used by

Minimality and closure laws of TC Proposition

Dependency tree · two levels

3 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

Marks, Set Theory, Berkeley edition — 7.5 p.34; Weiss p.97. (standard reference, not scraped)