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Definition df-dif 2947
Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union  ( A  u.  B ) (df-un 2949) and intersection  ( A  i^i  B ) (df-in 2951). Several notations are used in the literature; we chose the  \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " A excludes  B " to mean  A  \  B. We will use " B is removed from  A " to mean  A  \  { B } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
df-dif  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
Distinct variable groups:    x, A    x, B

Detailed syntax breakdown of Definition df-dif
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
31, 2cdif 2941 . 2  class  ( A 
\  B )
4 vx . . . . . 6  setvar  x
54cv 1258 . . . . 5  class  x
65, 1wcel 1409 . . . 4  wff  x  e.  A
75, 2wcel 1409 . . . . 5  wff  x  e.  B
87wn 3 . . . 4  wff  -.  x  e.  B
96, 8wa 101 . . 3  wff  ( x  e.  A  /\  -.  x  e.  B )
109, 4cab 2042 . 2  class  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }
113, 10wceq 1259 1  wff  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
Colors of variables: wff set class
This definition is referenced by:  dfdif2  2953  eldif  2954  bdcdif  10354
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