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Theorem brdif 4445
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 3424 . 2  |-  ( <. A ,  B >.  e.  ( R  \  S
)  <->  ( <. A ,  B >.  e.  R  /\  -.  <. A ,  B >.  e.  S ) )
2 df-br 4396 . 2  |-  ( A ( R  \  S
) B  <->  <. A ,  B >.  e.  ( R 
\  S ) )
3 df-br 4396 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4396 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
54notbii 294 . . 3  |-  ( -.  A S B  <->  -.  <. A ,  B >.  e.  S )
63, 5anbi12i 695 . 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 367    e. wcel 1842    \ cdif 3411   <.cop 3978   class class class wbr 4395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-dif 3417  df-br 4396
This theorem is referenced by:  fndmdif  5969  isocnv3  6211  brdifun  7375  dfsup2OLD  7937  dflt2  11407  pltval  15914  ltgov  24367  opeldifid  27892  qtophaus  28292  dftr6  29963  dffr5  29966  fundmpss  29980  brsset  30227  dfon3  30230  brtxpsd2  30233  dffun10  30252  elfuns  30253  dfrecs2  30288  dfrdg4  30289  dfint3  30290  brub  30292  broutsideof  30459  frege124d  35740
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