ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-06 (claude-opus-5)

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.

$(\varnothing,\varnothing) = \{\{\varnothing\}\}$ and $(\varnothing,\{\varnothing\}) = \{\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$ , with the characterising property checked on them

Example

Unfolding the Kuratowski definition at the two smallest sets gives

$(\varnothing,\varnothing) = \{\{\varnothing\}\}, \qquad (\varnothing,\{\varnothing\}) = \{\{\varnothing\},\{\varnothing,\{\varnothing\}\}\},$

and these two sets are distinct, as the characterising property requires, since $\varnothing \neq \{\varnothing\}$ .

Facts & Assumptions

Given: the sets $\varnothing$ and $\{\varnothing\}$ .

[L1]

$(a,b) := \{\{a\},\{a,b\}\}$ (The Kuratowski ordered pair $(a,b) := \{\{a\},\{a,b\}\}$ ).

[L2]

$\{x,y\}$ is the set whose elements are exactly $x$ and $y$ , and $\{x\} := \{x,x\}$ (The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$ ).

[L3]

$(a,b) = (c,d)$ if and only if $a = c$ and $b = d$ ( $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$ ).

[L4]

There is exactly one set with no elements (There is exactly one set with no elements, written $\varnothing$ ).

[L5]

If every $z$ satisfies $z \in x$ if and only if $z \in y$ , then $x = y$ (The Axiom of Extensionality: $\forall x\,\forall y\,(\forall z\,(z \in x \leftrightarrow z \in y) \to x = y)$ ).

Verification

technique · direct

1.1

$\varnothing \neq \{\varnothing\}$ , because $\{\varnothing\}$ has $\varnothing$ as an element and the empty set has none.

L2L4

1.2

$(\varnothing,\varnothing) = \{\{\varnothing\},\{\varnothing,\varnothing\}\}$ by the definition, and $\{\varnothing,\varnothing\} = \{\varnothing\}$ , so both members of the outer pair are $\{\varnothing\}$ and the outer pair is $\{\{\varnothing\}\}$ .

L1L2L5

1.3

$(\varnothing,\{\varnothing\}) = \{\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$ directly by the definition, with no collapse, since $\varnothing \neq \{\varnothing\}$ keeps the inner pair a set with two elements.

L1L2

2.1

The two pairs are distinct: their first coordinates agree but their second coordinates are $\varnothing$ and $\{\varnothing\}$ , which differ, so the characterising property forbids equality. This is visible in the computed sets as well, since $\{\varnothing,\{\varnothing\}\}$ is an element of the second and not of the first.

L3L5step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 5 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

Ordered pair (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), Def. 1 and Lemma 1 (standard reference, not scraped)
C. Wilson, A Brief Introduction to ZFC (Chicago REU 2016), Def. 2.6 (standard reference, not scraped)

(∅,∅)={{∅}}(\varnothing,\varnothing) = \{\{\varnothing\}\}(∅,∅)={{∅}} and (∅,{∅})={{∅},{∅,{∅}}}(\varnothing,\{\varnothing\}) = \{\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}(∅,{∅})={{∅},{∅,{∅}}}, with the characterising property checked on them