CorollaryStatement: Literature-sourcedProof: 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.

The universe is the class union of its stages

Statement

In ZF every set $x$ belongs to $V_{rank (x) + 1}$ . Consequently $V = ⋃_{α \in Ord} V_{α}$ in the class sense: every set lies in a stage. This is not a union indexed by a set of all ordinals.

Facts & Assumptions

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

[F1]

In ZF, for every set $x$ and ordinal $α$ , $x \in V_{α} ⟺ rank (x) < α, x \subseteq V_{α} ⟺ rank (x) \leq α .$ Thus $rank (x)$ is the least $α$ with $x \subseteq V_{α}$ , and $rank (x) = α$ iff $x \in V_{α + 1} ∖ V_{α}$ . (Rank characterizes hierarchy membership)

Proof

1.1

For $ρ = rank (x)$ , the ordinal inequality $ρ < ρ + 1$ and the membership characterization give $x \in V_{ρ + 1}$ .

2.1

Every member of a stage is a set, and step 1.1 gives a stage containing every set. Thus the two class descriptions agree. Neither statement asserts that their collection is a set.

step 1.1∎

Depends on

Rank characterizes hierarchy membership

Used by

V is a set False statement

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

Weiss, An Introduction to Set Theory (2014) — Theorem 40 p.100. (standard reference, not scraped)