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Theorem dfss4 3310
Description: Subclass defined in terms of class difference. See comments under dfun2 3311. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfss4  |-  ( A 
C_  B  <->  ( B  \  ( B  \  A
) )  =  A )

Proof of Theorem dfss4
StepHypRef Expression
1 sseqin2 3295 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2 eldif 3088 . . . . . . 7  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
32notbii 289 . . . . . 6  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
43anbi2i 678 . . . . 5  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 elin 3266 . . . . . 6  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  x  e.  A ) )
6 abai 773 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  <->  ( x  e.  B  /\  ( x  e.  B  ->  x  e.  A ) ) )
7 iman 415 . . . . . . 7  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 678 . . . . . 6  |-  ( ( x  e.  B  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
95, 6, 83bitri 264 . . . . 5  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9bitr4i 245 . . . 4  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  ( B  i^i  A ) )
1110difeqri 3213 . . 3  |-  ( B 
\  ( B  \  A ) )  =  ( B  i^i  A
)
1211eqeq1i 2260 . 2  |-  ( ( B  \  ( B 
\  A ) )  =  A  <->  ( B  i^i  A )  =  A )
131, 12bitr4i 245 1  |-  ( A 
C_  B  <->  ( B  \  ( B  \  A
) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3075    i^i cin 3077    C_ wss 3078
This theorem is referenced by:  dfin4  3316  sorpsscmpl  6140  sbthlem3  6858  fin23lem7  7826  fin23lem11  7827  compsscnvlem  7880  compssiso  7884  isf34lem4  7887  efgmnvl  14858  isopn2  16601  iincld  16608  iuncld  16614  clsval2  16619  ntrval2  16620  ntrdif  16621  clsdif  16622  cmclsopn  16631  cmntrcld  16632  opncldf1  16653  indiscld  16660  mretopd  16661  restcld  16735  pnrmopn  16903  conndisj  16974  hausllycmp  17052  kqcldsat  17256  filufint  17447  cfinufil  17455  ufilen  17457  alexsublem  17570  bcth3  18585  inmbl  18731  iccmbl  18755  mbfimaicc  18820  i1fd  18868  itgss3  19001  kur14lem4  22911  cldbnd  25410  clsun  25412  fdc  25621  frlmlbs  26415
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089
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