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Metamath Proof Explorer

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 i^i B ) = { x | ( x e. A /\ x e. B ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 
 |-  A
1 cB 
 |-  B
2 0 1 cin 
 |-  ( A i^i B )
3 vx 
 |-  x
4 3 cv 
 |-  x
5 4 0 wcel 
 |-  x e. A
6 4 1 wcel 
 |-  x e. B
7 5 6 wa 
 |-  ( x e. A /\ x e. B )
8 7 3 cab 
 |-  { x | ( x e. A /\ x e. B ) }
9 2 8 wceq 
 |-  ( A i^i B ) = { x | ( x e. A /\ x e. B ) }