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Theorem inssdif0 3811
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 3601 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 323 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 iman 422 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  <->  -.  (
( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C ) )
42, 3bitri 249 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3399 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 692 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3601 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 647 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 278 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9xchbinx 308 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1110albii 1648 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3406 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3727 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133bitr4i 277 1  |-  ( ( A  i^i  B ) 
C_  C  <->  ( A  i^i  ( B  \  C
) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397    = wceq 1399    e. wcel 1826    \ cdif 3386    i^i cin 3388    C_ wss 3389   (/)c0 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403  df-nul 3712
This theorem is referenced by:  disjdif  3816  inf3lem3  7961  ssfin4  8603  isnrm2  19945  1stccnp  20048  llycmpkgen2  20136  ufileu  20505  fclscf  20611  flimfnfcls  20614  inindif  27532  opnbnd  30309  diophrw  30857  setindtr  31132
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