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Theorem inssdif0 3534
Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
inssdif0  |-  ( ( A  i^i  B ) 
C_  C  <->  ( A  i^i  ( B  \  C
) )  =  (/) )

Proof of Theorem inssdif0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3371 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 315 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 iman 413 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  <->  -.  (
( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C ) )
42, 3bitri 240 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3175 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 675 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3371 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 630 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 269 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9xchbinx 301 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1110albii 1556 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3182 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3482 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133bitr4i 268 1  |-  ( ( A  i^i  B ) 
C_  C  <->  ( A  i^i  ( B  \  C
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  disjdif  3539  inf3lem3  7347  ssfin4  7952  isnrm2  17102  1stccnp  17204  llycmpkgen2  17261  ufileu  17630  fclscf  17736  flimfnfcls  17739  opnbnd  26346  diophrw  26941  setindtr  27220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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