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Theorem dfss4 3576
Description: Subclass defined in terms of class difference. See comments under dfun2 3577. (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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseqin2 3561 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2 eldif 3331 . . . . . . 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 677 . . . . 5  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 elin 3531 . . . . . 6  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  x  e.  A ) )
6 abai 772 . . . . . 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 677 . . . . . 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 3468 . . 3  |-  ( B 
\  ( B  \  A ) )  =  ( B  i^i  A
)
1211eqeq1i 2444 . 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 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3318    i^i cin 3320    C_ wss 3321
This theorem is referenced by:  dfin4  3582  sorpsscmpl  6534  sbthlem3  7220  fin23lem7  8197  fin23lem11  8198  compsscnvlem  8251  compssiso  8255  isf34lem4  8258  efgmnvl  15347  isopn2  17097  iincld  17104  iuncld  17110  clsval2  17115  ntrval2  17116  ntrdif  17117  clsdif  17118  cmclsopn  17127  cmntrcld  17128  opncldf1  17149  indiscld  17156  mretopd  17157  restcld  17237  pnrmopn  17408  conndisj  17480  hausllycmp  17558  kqcldsat  17766  filufint  17953  cfinufil  17961  ufilen  17963  alexsublem  18076  bcth3  19285  inmbl  19437  iccmbl  19461  mbfimaicc  19526  i1fd  19574  itgss3  19707  iundifdifd  24013  iundifdif  24014  cldssbrsiga  24542  kur14lem4  24896  mblfinlem3  26246  mblfinlem4  26247  ismblfin  26248  itg2addnclem  26257  cldbnd  26330  clsun  26332  fdc  26450  frlmlbs  27227  frgrawopreg2  28441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-dif 3324  df-in 3328  df-ss 3335
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