ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

${\emptyset} \times {\emptyset, {\emptyset}}$ listed in full, together with the inclusion in $P (P (A \cup B))$ that makes it a set

Example

Put $A : = {\emptyset}$ and $B : = {\emptyset, {\emptyset}}$ . Then

$A \times B = {{{\emptyset}}, {{\emptyset}, {\emptyset, {\emptyset}}}},$

a set with two elements, namely the pairs $(\emptyset, \emptyset)$ and $(\emptyset, {\emptyset})$ . Here $A \cup B = B$ , and both elements lie in $P (P (B))$ , which is the ambient set the product is separated inside.

Facts & Assumptions

Given: $A : = {\emptyset}$ and $B : = {\emptyset, {\emptyset}}$ .

[L1]

$z \in A \times B$ holds if and only if $z = (a, b)$ for some $a \in A$ and some $b \in B$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ).

[L2]

If $a \in A$ and $b \in B$ , then $(a, b) \in P (P (A \cup B))$ (If $a \in A$ and $b \in B$ then $(a, b) \in P (P (A \cup B))$ ).

[L3]

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

[L4]

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

[L5]

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ( $⋃ \emptyset = \emptyset$ , $⋃ {a} = a$ , $⋃ {a, b} = a \cup b$ , $⋂ {a} = a$ , and $⋂ {a, b} = a \cap b$ ).

[L6]

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

[L7]

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

[L8]

$z \in P (x)$ holds if and only if $z \subseteq x$ (The power set $P (x) = {z : z \subseteq x}$ ).

[L9]

$x \subseteq y$ means that every element of $x$ is an element of $y$ (Subset $x \subseteq y$ , proper subset $x ⊊ y$ , and the separation notation ${z \in x : φ (z)}$ ).

[L10]

$a \cup b : = ⋃ {a, b}$ (The union $⋃ x$ of a set, and the binary union $a \cup b : = ⋃ {a, b}$ ).

Verification

technique · direct

1.1

$A \cup B = B$ : an element of $A \cup B$ is $\emptyset$ or is an element of $B$ , and $\emptyset \in B$ , so the two sets have the same elements.

L4L5L6L7L10

1.2

The only element of $A$ is $\emptyset$ , and the elements of $B$ are $\emptyset$ and ${\emptyset}$ , so the pairs with first coordinate in $A$ and second in $B$ are exactly $(\emptyset, \emptyset)$ and $(\emptyset, {\emptyset})$ ; unfolding the definition of the ordered pair, these are ${{\emptyset}}$ and ${{\emptyset}, {\emptyset, {\emptyset}}}$ .

L1L3L4L6L7

2.1

Hence $A \times B$ has exactly those two elements, and they are distinct because ${\emptyset, {\emptyset}}$ belongs to the second and not to the first.

L1L4L6step 1.2

2.2

Both elements lie in $P (P (B))$ : this is the general fact applied with $A \cup B = B$ , and it is also visible directly, since each is a set of subsets of $B$ .

L2L8L9step 1.1step 1.2

3.1

The product is listed in full and its two elements are exhibited inside the ambient double power set that makes the separation legitimate.

step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Cartesian product (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), Def. 10 (standard reference, not scraped)
C. Wilson, A Brief Introduction to ZFC (Chicago REU 2016), Def. 2.8 (standard reference, not scraped)

{∅}×{∅,{∅}}{∅}×{∅,{∅}} listed in full, together with the inclusion in P(P(A∪B))P(P(A∪B)) that makes it a set