Definition df-xp

Description: Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of Quine p. 64. For example, ( { 1 , 5 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ) ( ex-xp ). Another example is that the set of rational numbers is defined in df-q using the Cartesian product ( ZZ X. NN ) ; the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994)

	Ref	Expression
Assertion	df-xp	$$⊢A\times B=\left\{⟨x,y⟩|\left(x\in A\wedge y\in B\right)\right\}$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	$$classA$$
1		cB	$$classB$$
2	0 1	cxp	$$class\left(A\times B\right)$$
3		vx	$$setvarx$$
4		vy	$$setvary$$
5	3	cv	$$setvarx$$
6	5 0	wcel	$$wffx\in A$$
7	4	cv	$$setvary$$
8	7 1	wcel	$$wffy\in B$$
9	6 8	wa	$$wff\left(x\in A\wedge y\in B\right)$$
10	9 3 4	copab	$$class\left\{⟨x,y⟩|\left(x\in A\wedge y\in B\right)\right\}$$
11	2 10	wceq	$$wffA\times B=\left\{⟨x,y⟩|\left(x\in A\wedge y\in B\right)\right\}$$