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Definition df-dif 3097
Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example,  ( { 1 ,  3 }  \  { 1 ,  8 } )  =  {
3 } (ex-dif 20718). Contrast this operation with union  ( A  u.  B
) (df-un 3099) and intersection  ( A  i^i  B ) (df-in 3101). 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 3091 . 2  class  ( A 
\  B )
4 vx . . . . . 6  set  x
54cv 1618 . . . . 5  class  x
65, 1wcel 1621 . . . 4  wff  x  e.  A
75, 2wcel 1621 . . . . 5  wff  x  e.  B
87wn 5 . . . 4  wff  -.  x  e.  B
96, 8wa 360 . . 3  wff  ( x  e.  A  /\  -.  x  e.  B )
109, 4cab 2242 . 2  class  { x  |  ( x  e.  A  /\  -.  x  e.  B ) }
113, 10wceq 1619 1  wff  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
Colors of variables: wff set class
This definition is referenced by:  dfdif2  3103  eldif  3104  ballotth  23022
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