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Theorem inssdif0 3845
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 3628 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 331 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 iman 430 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  <->  -.  (
( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C ) )
42, 3bitri 257 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3425 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 705 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3628 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 659 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 286 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9xchbinx 316 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1110albii 1701 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3432 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3758 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133bitr4i 285 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 189    /\ wa 375   A.wal 1452    = wceq 1454    e. wcel 1897    \ cdif 3412    i^i cin 3414    C_ wss 3415   (/)c0 3742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-ss 3429  df-nul 3743
This theorem is referenced by:  disjdif  3850  inf3lem3  8160  ssfin4  8765  isnrm2  20422  1stccnp  20525  llycmpkgen2  20613  ufileu  20982  fclscf  21088  flimfnfcls  21091  inindif  28198  opnbnd  31029  diophrw  35645  setindtr  35923
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