ExampleConstruction: AI-generatedVerification: 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.

For any indexed family the product $\prod_{i \in I} P (X_{i})$ contains the constant function with value $\emptyset$ , and $\prod_{i \in I} {i}$ has exactly one element

Example

Two families whose products can be shown nonempty by writing an element down, with no choice principle involved.

Let $(X_{i})_{i \in I}$ be any indexed family. Then $(P (X_{i}))_{i \in I}$ is an indexed family, and the constant function $c : = {(i, \emptyset) : i \in I}$ is an element of $\prod_{i \in I} P (X_{i})$ . So that product is nonempty for every $I$ and every family, even when some $X_{i}$ is empty.
Let $I$ be any set and let $({i})_{i \in I}$ be the family carrying the singleton of the index at each index. Then $\prod_{i \in I} {i} = {Δ_{I}}$ : its only element is the identity relation on $I$ .

Neither construction selects anything: in the first the value is the same set at every index, and in the second the value at $i$ is forced to be $i$ .

Facts & Assumptions

Given: an indexed family $(X_{i})_{i \in I}$ and a set $I$ .

[L1]

an element of $\prod_{i \in I} A_{i}$ is a function with domain $I$ that takes its value at each index inside the member carried by that index (The product $\prod_{i \in I} A_{i} : = {f : I \to ⋃_{i \in I} A_{i} ∣ f (i) \in A_{i} for every i \in I}$ ).

[L2]

An indexed family with index set $I$ is a function $A$ with $dom A = I$ (An indexed family $(A_{i})_{i \in I}$ is a function with domain $I$ ; ${A_{i} : i \in I}$ is its range).

[L3]

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

[L4]

(i) $\emptyset \subseteq x$ ( $\emptyset \subseteq x$ , $x \subseteq x$ , inclusion is transitive, and $x = y$ if and only if $x \subseteq y$ and $y \subseteq x$ ).

[L5]

A function is a relation $f$ such that $(a, b) \in f$ and $(a, c) \in f$ imply $b = c$ (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain).

[L6]

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

[L7]

$Δ_{A}$ is a function with $dom Δ_{A} = A$ and $Δ_{A} (a) = a$ for every $a \in A$ (If $f$ and $g$ are functions then $g \circ f$ is a function with domain $f^{- 1} [dom g]$ and $(g \circ f) (x) = g (f (x))$ there; $Δ_{A}$ is a function with $Δ_{A} (a) = a$ ; and $f \circ Δ_{A} = f = Δ_{B} \circ f$ for $f : A \to B$ ).

[L8]

$(a, b) \in Δ_{A}$ holds if and only if $a = b$ and $a \in A$ (The identity relation $Δ_{A} = {(a, b) \in A \times A : a = b}$ and the membership relation $\in_{A} = {(a, b) \in A \times A : a \in b}$ ).

[L9]

${x} : = {x, x}$ , the singleton of $x$ , is the set whose only element is $x$ (The unordered pair ${x, y}$ and the singleton ${x} = {x, x}$ ).

[L10]

if $I = \emptyset$ then $\prod_{i \in I} A_{i} = {\emptyset}$ ( $\prod_{i \in \emptyset} A_{i} = {\emptyset}$ ; if $A_{j} = \emptyset$ for some $j \in I$ then $\prod_{i \in I} A_{i} = \emptyset$ ; and for $I = {j}$ the evaluation $f \mapsto f (j)$ is a bijection $\prod_{i \in I} A_{i} \to A_{j}$ ).

[L11]

For any parameters $\overset{ˉ}{p}$ and any set $x$ , there is a set $y$ whose elements are exactly the elements $z$ of $x$ for which $φ (z, \overset{ˉ}{p})$ holds (The Axiom Schema of Separation: for each formula $φ$ , $\forall \overset{ˉ}{p} \forall x \exists y \forall z (z \in y \leftrightarrow (z \in x \land φ (z, \overset{ˉ}{p})))$ ).

[L12]

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

[L13]

$dom R : = {a : \exists b (a, b) \in R}, ran R : = {b : \exists a (a, b) \in R}$ (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L14]

$z \in ⋃_{i \in I} A_{i}$ holds if and only if $z \in A_{i}$ for some $i \in I$ ( $⋃_{i \in I} A_{i} : = ⋃ {A_{i} : i \in I}$ , and $⋂_{i \in I} A_{i} : = ⋂ {A_{i} : i \in I}$ for $I \neq \emptyset$ ).

Verification

technique · direct

1.1L2L3L5L11L12L13L14

$(P (X_{i}))_{i \in I}$ is an indexed family: separating inside $I \times P (P (⋃_{i \in I} X_{i}))$ with the formula $\exists i \exists w (z = (i, w) \land i \in I \land w = P (X_{i}))$ gives a set, which is single valued and has domain $I$ ; the ambient set contains each $P (X_{i})$ , since every subset of $X_{i}$ is a subset of $⋃_{i \in I} X_{i}$ .

1.2L2L3L5L9L11L12L13

Similarly $({i})_{i \in I}$ is an indexed family, obtained by separating inside $I \times P (I)$ , since ${i} \subseteq I$ for $i \in I$ .

2.1L1L3L4L5L6L9L11L12L13step 1.1

The constant function $c : = {(i, \emptyset) : i \in I}$ is a set, by separating inside $I \times {\emptyset}$ ; it is single valued, has domain $I$ , and $c (i) = \emptyset$ for every $i \in I$ . Since $\emptyset \subseteq X_{i}$ , we have $\emptyset \in P (X_{i})$ for every $i$ , so $c$ lies in $\prod_{i \in I} P (X_{i})$ and that product is nonempty.

2.2L1L5L7L8L9step 1.2

$Δ_{I}$ is a function with domain $I$ and $Δ_{I} (i) = i$ , and $i$ is the only element of ${i}$ , so $Δ_{I} \in \prod_{i \in I} {i}$ . Conversely any $f$ in that product has domain $I$ and $f (i) \in {i}$ , hence $f (i) = i$ for every $i \in I$ , so $f$ and $Δ_{I}$ are functions with the same domain agreeing everywhere and are equal.

3.1L10step 2.1step 2.2∎

Both products are therefore nonempty, and the second has exactly one element; when $I = \emptyset$ both statements agree with the general computation of the empty product, whose single element is the empty function.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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

Axiom of choice (Wikipedia) (standard reference, not scraped)
Cartesian product (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), Def. 21 and Exercise 13 (standard reference, not scraped)

For any indexed family the product ∏i∈IP(Xi)∏i∈I​P(Xi​) contains the constant function with value ∅∅, and ∏i∈I{i}∏i∈I​{i} has exactly one element