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Theorem brdif 4490
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 3479 . 2  |-  ( <. A ,  B >.  e.  ( R  \  S
)  <->  ( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
2 df-br 4441 . 2  |-  ( A ( R  \  S
) B  <->  <. A ,  B >.  e.  ( R 
\  S ) )
3 df-br 4441 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4441 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
54notbii 296 . . 3  |-  ( -.  A S B  <->  -.  <. A ,  B >.  e.  S )
63, 5anbi12i 697 . 2  |-  ( ( A R B  /\  -.  A S B )  <-> 
( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
71, 2, 63bitr4i 277 1  |-  ( A ( R  \  S
) B  <->  ( A R B  /\  -.  A S B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    e. wcel 1762    \ cdif 3466   <.cop 4026   class class class wbr 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-dif 3472  df-br 4441
This theorem is referenced by:  fndmdif  5976  isocnv3  6207  brdifun  7328  dfsup2OLD  7892  dflt2  11343  pltval  15436  ltgov  23703  opeldifid  27115  qtophaus  27623  dftr6  28742  dffr5  28745  fundmpss  28759  brsset  29102  dfon3  29105  brtxpsd2  29108  dffun10  29127  elfuns  29128  dfrdg4  29163  dfint3  29165  brub  29167  broutsideof  29334
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