MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-in Unicode version

Definition df-in 3120
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 20741). Contrast this operation with union  ( A  u.  B
) (df-un 3118) and difference  ( A  \  B ) (df-dif 3116). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3366 and dfin4 3370. For intersection defined in terms of union, see dfin3 3369. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
df-in  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Detailed syntax breakdown of Definition df-in
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
31, 2cin 3112 . 2  class  ( A  i^i  B )
4 vx . . . . . 6  set  x
54cv 1618 . . . . 5  class  x
65, 1wcel 1621 . . . 4  wff  x  e.  A
75, 2wcel 1621 . . . 4  wff  x  e.  B
86, 7wa 360 . . 3  wff  ( x  e.  A  /\  x  e.  B )
98, 4cab 2242 . 2  class  { x  |  ( x  e.  A  /\  x  e.  B ) }
103, 9wceq 1619 1  wff  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
Colors of variables: wff set class
This definition is referenced by:  dfin5  3121  dfss2  3130  elin  3319  disj  3456  iinxprg  3939  csbingVD  27694
  Copyright terms: Public domain W3C validator