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Theorem brdif 4071
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 3162 . 2  |-  ( <. A ,  B >.  e.  ( R  \  S
)  <->  ( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
2 df-br 4024 . 2  |-  ( A ( R  \  S
) B  <->  <. A ,  B >.  e.  ( R 
\  S ) )
3 df-br 4024 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4024 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
54notbii 287 . . 3  |-  ( -.  A S B  <->  -.  <. A ,  B >.  e.  S )
63, 5anbi12i 678 . 2  |-  ( ( A R B  /\  -.  A S B )  <-> 
( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
71, 2, 63bitr4i 268 1  |-  ( A ( R  \  S
) B  <->  ( A R B  /\  -.  A S B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    e. wcel 1684    \ cdif 3149   <.cop 3643   class class class wbr 4023
This theorem is referenced by:  fndmdif  5629  isocnv3  5829  brdifun  6687  dfsup2OLD  7196  dflt2  10482  pltval  14094  dftr6  24107  dffr5  24110  fundmpss  24122  brsset  24429  dfon3  24432  brtxpsd2  24435  dffun10  24453  elfuns  24454  dfrdg4  24488  broutsideof  24744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-br 4024
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