PropositionStatement: AI-adaptedProof: AI-generatedPipeline-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.

Minimality and closure laws of TC

Statement

For every set $a$ , $TC (a)$ is transitive, contains $a$ as a subset, and is contained in every transitive set $T$ with $a \subseteq T$ . Moreover $a \subseteq b$ implies $TC (a) \subseteq TC (b)$ , and $TC (TC (a)) = TC (a)$ . In particular $a \in TC ({a})$ .

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

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: lem-finite-predecessor-closure-is-a-set. (Transitive closure of a set)

Proof

1.1F1

The stage zero inclusion gives $a \subseteq TC (a)$ . If $z \in y \in TC (a)$ , choose $n$ with $y \in a_{n}$ ; then $z \in ⋃ a_{n} \subseteq a_{n + 1}$ . Thus the union is transitive. Equivalently, transitivity of a set $T$ is the elementary condition $⋃ T \subseteq T$ .

2.1F1step 1.1

If $a \subseteq T$ and $T$ is transitive, induction gives $a_{n} \subseteq T$ : the successor follows from $⋃ a_{n} \subseteq ⋃ T \subseteq T$ . Union over $n$ proves minimality.

3.1step 1.1step 2.1∎

For $a \subseteq b$ , the transitive set $TC (b)$ contains $a$ , so minimality proves monotonicity. Applying minimality with $T = TC (a)$ gives one idempotence inclusion and step 1.1 gives the other. Finally $a \in {a} \subseteq TC ({a})$ .

Depends on

Transitive closure of a set

Used by

Well-founded pointed graphs have unique decorations Corollary
Grothendieck universe closure convention Definition
Hereditary size and Hₖappa Definition
Ordinal definability and HOD Definition
Equivalent forms of Foundation Theorem

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

Marks, Set Theory, Berkeley edition — 7.6 and 7.9 pp.34–35. (standard reference, not scraped)