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Theorem brdif 3868
Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
Assertion
Ref Expression
brdif  |-  ( A ( R  \  S
) B  <->  ( A R B  /\  -.  A S B ) )

Proof of Theorem brdif
StepHypRef Expression
1 eldif 2997 . 2  |-  ( <. A ,  B >.  e.  ( R  \  S
)  <->  ( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
2 df-br 3821 . 2  |-  ( A ( R  \  S
) B  <->  <. A ,  B >.  e.  ( R 
\  S ) )
3 df-br 3821 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 3821 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
54notbii 627 . . 3  |-  ( -.  A S B  <->  -.  <. A ,  B >.  e.  S )
63, 5anbi12i 448 . 2  |-  ( ( A R B  /\  -.  A S B )  <-> 
( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
71, 2, 63bitr4i 210 1  |-  ( A ( R  \  S
) B  <->  ( A R B  /\  -.  A S B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    e. wcel 1436    \ cdif 2985   <.cop 3434   class class class wbr 3820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-br 3821
This theorem is referenced by:  fndmdif  5367  brdifun  6271
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