PropositionStatement: 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.

$A \times (B \cup C) = (A \times B) \cup (A \times C)$ , $A \times (B \cap C) = (A \times B) \cap (A \times C)$ , $A \times (B ∖ C) = (A \times B) ∖ (A \times C)$ , $(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)$ ; $A \times B = \emptyset$ if and only if $A = \emptyset$ or $B = \emptyset$ ; and for nonempty $A$ and $B$ , $A \times B \subseteq C \times D$ if and only if $A \subseteq C$ and $B \subseteq D$

Statement

For all sets $A$ , $B$ , $C$ , $D$ :

(i) $A \times (B \cup C) = (A \times B) \cup (A \times C)$ ;
(ii) $A \times (B \cap C) = (A \times B) \cap (A \times C)$ ;
(iii) $A \times (B ∖ C) = (A \times B) ∖ (A \times C)$ ;
(iv) $(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)$ ;
(v) $A \times B = \emptyset$ if and only if $A = \emptyset$ or $B = \emptyset$ ;
(vi) if $A \neq \emptyset$ and $B \neq \emptyset$ , then $A \times B \subseteq C \times D$ if and only if $A \subseteq C$ and $B \subseteq D$ .

Facts & Assumptions

Given: sets $A$ , $B$ , $C$ , $D$ .

[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]

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

[L3]

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

[L4]

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

[L5]

$z \in a ∖ b$ 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 $X$ ).

[L6]

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

[L7]

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

[L8]

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

[L9]

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

[L10]

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

[L11]

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

Proof

technique · direct

1.1

Membership criterion: for all sets $a$ and $b$ , $(a, b) \in A \times B$ holds if and only if $a \in A$ and $b \in B$ . Indeed, an element of $A \times B$ is a pair $(a^{'}, b^{'})$ with $a^{'} \in A$ and $b^{'} \in B$ , and $(a, b) = (a^{'}, b^{'})$ forces $a = a^{'}$ and $b = b^{'}$ ; the converse is immediate from the description of $A \times B$ . Every element of a product is a pair, so it suffices in each identity below to compare pairs.

L1L2L9

2.1

Claim (i): $(a, t) \in A \times (B \cup C)$ holds exactly when $a \in A$ and $t \in B$ or $t \in C$ , that is, exactly when $(a, t) \in A \times B$ or $(a, t) \in A \times C$ .

L3L8L10step 1.1

2.2

Claim (ii): $(a, t) \in A \times (B \cap C)$ holds exactly when $a \in A$ , $t \in B$ and $t \in C$ , that is, exactly when $(a, t) \in A \times B$ and $(a, t) \in A \times C$ .

L4L8L11step 1.1

2.3

Claim (iii): $(a, t) \in A \times (B ∖ C)$ holds exactly when $a \in A$ , $t \in B$ and $t \notin C$ . On the other side, $(a, t) \in (A \times B) ∖ (A \times C)$ holds exactly when $a \in A$ , $t \in B$ , and it is not the case that $a \in A$ and $t \in C$ ; given $a \in A$ , that last condition is $t \notin C$ .

L5L8step 1.1

2.4

Claim (iv): $(u, v) \in (A \cap B) \times (C \cap D)$ holds exactly when $u \in A$ , $u \in B$ , $v \in C$ and $v \in D$ , that is, exactly when $(u, v) \in A \times C$ and $(u, v) \in B \times D$ .

L4L8L11step 1.1

2.5

Claim (v): if $A = \emptyset$ or $B = \emptyset$ then no pair satisfies the membership criterion, so $A \times B$ has no elements and equals $\emptyset$ ; conversely if both are nonempty, fix $a \in A$ and $b \in B$ , and then $(a, b) \in A \times B$ .

L7step 1.1

2.6

Claim (vi): assume $A \neq \emptyset$ and $B \neq \emptyset$ . If $A \times B \subseteq C \times D$ , fix $b_{0} \in B$ ; for any $a \in A$ the pair $(a, b_{0})$ lies in $A \times B$ , hence in $C \times D$ , so $a \in C$ , and $A \subseteq C$ follows; fixing $a_{0} \in A$ and running the same argument on the second coordinate gives $B \subseteq D$ . Conversely, if $A \subseteq C$ and $B \subseteq D$ , then any $(a, b) \in A \times B$ has $a \in C$ and $b \in D$ , so it lies in $C \times D$ .

L6L7step 1.1

3.1

Claims (i) to (vi) are established, which is the statement.

step 2.1step 2.2step 2.3step 2.4step 2.5step 2.6∎

Depends on

Used by

Dependency tree · two levels

19 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), §2.1 (standard reference, not scraped)
Algebra of sets (Wikipedia) (standard reference, not scraped)