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Theorem dfss4 3378
Description: Subclass defined in terms of class difference. See comments under dfun2 3379. (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 3363 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2 eldif 3137 . . . . . . 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 3333 . . . . . 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 3271 . . 3  |-  ( B 
\  ( B  \  A ) )  =  ( B  i^i  A
)
1211eqeq1i 2265 . 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 3124    i^i cin 3126    C_ wss 3127
This theorem is referenced by:  dfin4  3384  sorpsscmpl  6222  sbthlem3  6941  fin23lem7  7910  fin23lem11  7911  compsscnvlem  7964  compssiso  7968  isf34lem4  7971  efgmnvl  14985  isopn2  16731  iincld  16738  iuncld  16744  clsval2  16749  ntrval2  16750  ntrdif  16751  clsdif  16752  cmclsopn  16761  cmntrcld  16762  opncldf1  16783  indiscld  16790  mretopd  16791  restcld  16865  pnrmopn  17033  conndisj  17104  hausllycmp  17182  kqcldsat  17386  filufint  17577  cfinufil  17585  ufilen  17587  alexsublem  17700  bcth3  18715  inmbl  18861  iccmbl  18885  mbfimaicc  18950  i1fd  18998  itgss3  19131  kur14lem4  23112  cldbnd  25611  clsun  25613  fdc  25822  frlmlbs  26616
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141
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