Definition df-in

Description: Define the intersection of two classes. Definition 5.6 of TakeutiZaring p. 16. For example, ( { 1 , 3 } i^i { 1 , 8 } ) = { 1 } ( ex-in ). Contrast this operation with union ( A u. B ) ( df-un ) and difference ( A \ B ) ( df-dif ). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 and dfin4 . For intersection defined in terms of union, see dfin3 . (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	df-in	$$⊢A\cap B=\left\{x|\left(x\in A\wedge x\in B\right)\right\}$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	$$classA$$
1		cB	$$classB$$
2	0 1	cin	$$class\left(A\cap B\right)$$
3		vx	$$setvarx$$
4	3	cv	$$setvarx$$
5	4 0	wcel	$$wffx\in A$$
6	4 1	wcel	$$wffx\in B$$
7	5 6	wa	$$wff\left(x\in A\wedge x\in B\right)$$
8	7 3	cab	$$class\left\{x|\left(x\in A\wedge x\in B\right)\right\}$$
9	2 8	wceq	$$wffA\cap B=\left\{x|\left(x\in A\wedge x\in B\right)\right\}$$