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Theorem dfss4 3364
Description: Subclass defined in terms of class difference. See comments under dfun2 3365. (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 3349 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2 eldif 3123 . . . . . . 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 3319 . . . . . 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 3257 . . 3  |-  ( B 
\  ( B  \  A ) )  =  ( B  i^i  A
)
1211eqeq1i 2263 . 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 3110    i^i cin 3112    C_ wss 3113
This theorem is referenced by:  dfin4  3370  sorpsscmpl  6208  sbthlem3  6927  fin23lem7  7896  fin23lem11  7897  compsscnvlem  7950  compssiso  7954  isf34lem4  7957  efgmnvl  14971  isopn2  16717  iincld  16724  iuncld  16730  clsval2  16735  ntrval2  16736  ntrdif  16737  clsdif  16738  cmclsopn  16747  cmntrcld  16748  opncldf1  16769  indiscld  16776  mretopd  16777  restcld  16851  pnrmopn  17019  conndisj  17090  hausllycmp  17168  kqcldsat  17372  filufint  17563  cfinufil  17571  ufilen  17573  alexsublem  17686  bcth3  18701  inmbl  18847  iccmbl  18871  mbfimaicc  18936  i1fd  18984  itgss3  19117  kur14lem4  23098  cldbnd  25597  clsun  25599  fdc  25808  frlmlbs  26602
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 2237
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-dif 3116  df-in 3120  df-ss 3127
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