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Theorem brdif 4228
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 3298 . 2  |-  ( <. A ,  B >.  e.  ( R  \  S
)  <->  ( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
2 df-br 4181 . 2  |-  ( A ( R  \  S
) B  <->  <. A ,  B >.  e.  ( R 
\  S ) )
3 df-br 4181 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4181 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
54notbii 288 . . 3  |-  ( -.  A S B  <->  -.  <. A ,  B >.  e.  S )
63, 5anbi12i 679 . 2  |-  ( ( A R B  /\  -.  A S B )  <-> 
( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
71, 2, 63bitr4i 269 1  |-  ( A ( R  \  S
) B  <->  ( A R B  /\  -.  A S B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    e. wcel 1721    \ cdif 3285   <.cop 3785   class class class wbr 4180
This theorem is referenced by:  fndmdif  5801  isocnv3  6019  brdifun  6899  dfsup2OLD  7414  dflt2  10705  pltval  14380  dftr6  25329  dffr5  25332  fundmpss  25344  brsset  25651  dfon3  25654  brtxpsd2  25657  dffun10  25675  elfuns  25676  dfrdg4  25711  broutsideof  25967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-dif 3291  df-br 4181
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