TheoremStatement: Literature-sourcedProof: 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.

$X \setminus (a \cup b) = (X \setminus a) \cap (X \setminus b)$ and $X \setminus (a \cap b) = (X \setminus a) \cup (X \setminus b)$ ; and for a nonempty set $F$ , $X \setminus \bigcup F = \bigcap \{\, X \setminus a : a \in F \,\}$ and $X \setminus \bigcap F = \bigcup \{\, X \setminus a : a \in F \,\}$

Statement

For all sets $X$ , $a$ and $b$ ,

$X \setminus (a \cup b) = (X \setminus a) \cap (X \setminus b), \qquad X \setminus (a \cap b) = (X \setminus a) \cup (X \setminus b).$

Let $F$ be a set with $F \neq \varnothing$ . Then $\{\, X \setminus a : a \in F \,\}$ is a nonempty set and

$X \setminus \bigcup F = \bigcap \{\, X \setminus a : a \in F \,\}, \qquad X \setminus \bigcap F = \bigcup \{\, X \setminus a : a \in F \,\}.$

Facts & Assumptions

Given: sets $X$ , $a$ , $b$ , and a set $F$ with $F \neq \varnothing$ where the family forms are concerned.

[L1]

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

[L2]

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ( $\bigcup \varnothing = \varnothing$ , $\bigcup \{a\} = a$ , $\bigcup \{a,b\} = a \cup b$ , $\bigcap \{a\} = a$ , and $\bigcap \{a,b\} = a \cap b$ ).

[L3]

$z \in a \cap b$ holds if and only if $z \in a$ and $z \in b$ ( $\bigcup \varnothing = \varnothing$ , $\bigcup \{a\} = a$ , $\bigcup \{a,b\} = a \cup b$ , $\bigcap \{a\} = a$ , and $\bigcap \{a,b\} = a \cap b$ ).

[L4]

$\bigcup x$ is the set whose elements are exactly the elements of the elements of $x$ (The union $\bigcup x$ of a set, and the binary union $a \cup b := \bigcup \{a,b\}$ ).

[L5]

For $x \neq \varnothing$ , $\bigcap x$ is the set whose elements are exactly the sets belonging to every element of $x$ (The intersection $\bigcap x$ of a nonempty set, the binary intersection $a \cap b := \bigcap\{a,b\}$ , and disjointness).

[L6]

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

[L7]

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

[L8]

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

[L9]

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

Proof

technique · direct

1.1

Every $X \setminus a$ is a subset of $X$ , so the class $\{\, X \setminus a : a \in F \,\}$ is the separated set $\{\, w \in \mathcal{P}(X) : \exists a\,(a \in F \wedge w = X \setminus a) \,\}$ , hence a set; and it has a member because $F$ does.

L1L6L7L8

1.2

$z \in X \setminus (a \cup b)$ holds exactly when $z \in X$ and neither $z \in a$ nor $z \in b$ , that is, exactly when $z \in X$ and $z \notin a$ , and $z \in X$ and $z \notin b$ ; this is $z \in (X \setminus a) \cap (X \setminus b)$ .

L1L2L3L9

1.3

$z \in X \setminus (a \cap b)$ holds exactly when $z \in X$ and not both $z \in a$ and $z \in b$ , that is, exactly when $z \in X$ and $z \notin a$ , or $z \in X$ and $z \notin b$ ; this is $z \in (X \setminus a) \cup (X \setminus b)$ .

L1L2L3L9

2.1

$z \in X \setminus \bigcup F$ holds exactly when $z \in X$ and no member $s$ of $F$ has $z \in s$ , that is, exactly when $z \in X \setminus a$ for every $a \in F$ ; the sets $X \setminus a$ with $a \in F$ are precisely the elements of $\{\, X \setminus a : a \in F \,\}$ , which is nonempty, so this is $z \in \bigcap \{\, X \setminus a : a \in F \,\}$ .

L1L4L5L9step 1.1

2.2

$z \in X \setminus \bigcap F$ holds exactly when $z \in X$ and $z$ fails to lie in some member $a$ of $F$ , that is, exactly when $z \in X \setminus a$ for some $a \in F$ ; this is $z \in \bigcup \{\, X \setminus a : a \in F \,\}$ .

L1L4L5L9step 1.1

3.1

The two finite laws and the two family laws are established, which is the statement.

step 1.2step 1.3step 2.1step 2.2∎

Depends on

Used by

For a nonempty index set I: B ∩ ⋃_i ∈ I Aᵢ = ⋃_i ∈ I (B ∩ Aᵢ), B ∪ ⋂_i ∈ I Aᵢ = ⋂_i ∈ I (B ∪ Aᵢ), X ∖ ⋃_i ∈ I Aᵢ = ⋂_i ∈ I (X ∖ Aᵢ), and X ∖ ⋂_i ∈ I Aᵢ = ⋃_i ∈ I (X ∖ Aᵢ) Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 6 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

De Morgan's laws (Wikipedia) (standard reference, not scraped)
Algebra of sets (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), §1.2 (standard reference, not scraped)