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Theorem dfss4 3403
Description: Subclass defined in terms of class difference. See comments under dfun2 3404. (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 3388 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2 eldif 3162 . . . . . . 7  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
32notbii 287 . . . . . 6  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
43anbi2i 675 . . . . 5  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 elin 3358 . . . . . 6  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  x  e.  A ) )
6 abai 770 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  <->  ( x  e.  B  /\  ( x  e.  B  ->  x  e.  A ) ) )
7 iman 413 . . . . . . 7  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 675 . . . . . 6  |-  ( ( x  e.  B  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
95, 6, 83bitri 262 . . . . 5  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9bitr4i 243 . . . 4  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  ( B  i^i  A ) )
1110difeqri 3296 . . 3  |-  ( B 
\  ( B  \  A ) )  =  ( B  i^i  A
)
1211eqeq1i 2290 . 2  |-  ( ( B  \  ( B 
\  A ) )  =  A  <->  ( B  i^i  A )  =  A )
131, 12bitr4i 243 1  |-  ( A 
C_  B  <->  ( B  \  ( B  \  A
) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152
This theorem is referenced by:  dfin4  3409  sorpsscmpl  6288  sbthlem3  6973  fin23lem7  7942  fin23lem11  7943  compsscnvlem  7996  compssiso  8000  isf34lem4  8003  efgmnvl  15023  isopn2  16769  iincld  16776  iuncld  16782  clsval2  16787  ntrval2  16788  ntrdif  16789  clsdif  16790  cmclsopn  16799  cmntrcld  16800  opncldf1  16821  indiscld  16828  mretopd  16829  restcld  16903  pnrmopn  17071  conndisj  17142  hausllycmp  17220  kqcldsat  17424  filufint  17615  cfinufil  17623  ufilen  17625  alexsublem  17738  bcth3  18753  inmbl  18899  iccmbl  18923  mbfimaicc  18988  i1fd  19036  itgss3  19169  iundifdifd  23159  iundifdif  23160  cldssbrsiga  23518  kur14lem4  23740  cldbnd  26244  clsun  26246  fdc  26455  frlmlbs  27249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166
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