CorollaryStatement: AI-adaptedProof: 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 a nonempty index set $I$ : $B \cap ⋃_{i \in I} A_{i} = ⋃_{i \in I} (B \cap A_{i})$ , $B \cup ⋂_{i \in I} A_{i} = ⋂_{i \in I} (B \cup A_{i})$ , $X ∖ ⋃_{i \in I} A_{i} = ⋂_{i \in I} (X ∖ A_{i})$ , and $X ∖ ⋂_{i \in I} A_{i} = ⋃_{i \in I} (X ∖ A_{i})$

Statement

Let $(A_{i})_{i \in I}$ be an indexed family with $I \neq \emptyset$ and let $B$ and $X$ be sets. Then $(B \cap A_{i})_{i \in I}$ , $(B \cup A_{i})_{i \in I}$ and $(X ∖ A_{i})_{i \in I}$ are indexed families with index set $I$ , and

$B \cap ⋃_{i \in I} A_{i} = ⋃_{i \in I} (B \cap A_{i}), B \cup ⋂_{i \in I} A_{i} = ⋂_{i \in I} (B \cup A_{i}),$

$X ∖ ⋃_{i \in I} A_{i} = ⋂_{i \in I} (X ∖ A_{i}), X ∖ ⋂_{i \in I} A_{i} = ⋃_{i \in I} (X ∖ A_{i}) .$

Facts & Assumptions

Given: an indexed family $(A_{i})_{i \in I}$ with $I \neq \emptyset$ , and sets $B$ and $X$ .

[L1]

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

[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]

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

[L4]

$a \cap ⋃ F = ⋃ {a \cap b : b \in F}, a \cup ⋂ F = ⋂ {a \cup b : b \in F}$ for a set $F$ with $F \neq \emptyset$ ( $a \cap (b \cup c) = (a \cap b) \cup (a \cap c)$ and $a \cup (b \cap c) = (a \cup b) \cap (a \cup c)$ ; and for a nonempty set $F$ , $a \cap ⋃ F = ⋃ {a \cap b : b \in F}$ and $a \cup ⋂ F = ⋂ {a \cup b : b \in F}$ ).

[L5]

$X ∖ ⋃ F = ⋂ {X ∖ a : a \in F}, X ∖ ⋂ F = ⋃ {X ∖ a : a \in F}$ for a set $F$ with $F \neq \emptyset$ ( $X ∖ (a \cup b) = (X ∖ a) \cap (X ∖ b)$ and $X ∖ (a \cap b) = (X ∖ a) \cup (X ∖ b)$ ; and for a nonempty set $F$ , $X ∖ ⋃ F = ⋂ {X ∖ a : a \in F}$ and $X ∖ ⋂ F = ⋃ {X ∖ a : a \in F}$ ).

[L6]

$z \in a ∖ b$ holds exactly when $z \in a$ and $z \notin b$ (The difference $a ∖ b$ , the symmetric difference $a △ b$ , and the complement $X ∖ a$ relative to a set $X$ ).

[L7]

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

[L8]

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

[L9]

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

[L10]

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 $φ$ , $\forall \overset{ˉ}{p} \forall x \exists y \forall z (z \in y \leftrightarrow (z \in x \land φ (z, \overset{ˉ}{p})))$ ).

[L11]

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

[L12]

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).

Proof

technique · direct

1.1

The three derived families exist. Each $B \cap A_{i}$ is a subset of $B$ , so separating inside $I \times P (B)$ with the formula $\exists i \exists w (z = (i, w) \land i \in I \land w = B \cap A_{i})$ gives a set; it is a function, since the value at each $i$ is determined, and its domain is $I$ . The same construction inside $I \times P (B \cup ⋃_{i \in I} A_{i})$ and inside $I \times P (X)$ gives $(B \cup A_{i})_{i \in I}$ and $(X ∖ A_{i})_{i \in I}$ .

L2L9L10L11L12

2.1

Write $F : = {A_{i} : i \in I}$ , which is $ran A$ and is nonempty because $I$ is. The range of $(B \cap A_{i})_{i \in I}$ is exactly ${B \cap b : b \in F}$ , since the values of the derived family are the sets $B \cap A_{i}$ with $i \in I$ and the elements of $F$ are exactly the $A_{i}$ with $i \in I$ ; the same holds for the other two derived families.

L2L3L8step 1.1

3.1

Substituting into the family laws for $F$ therefore gives all four identities: the indexed operations are by definition the primitive $⋃$ and $⋂$ applied to the range of the family concerned, and step 2.1 identifies those ranges with the sets appearing in the laws.

L1L4L5L6L7step 2.1

4.1

The derived families exist and the four identities hold, which is the statement.

step 1.1step 2.1step 3.1∎

Depends on

Used by

Open-dense and closed-nowhere-dense Baire forms are equivalent in ZF Lemma

Dependency tree · two levels

23 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

De Morgan's laws (Wikipedia) (standard reference, not scraped)
Indexed family (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), §2.3 (standard reference, not scraped)