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Theorem inssdif0 3894
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 3687 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 325 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 iman 424 . . . . 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 3486 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 694 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3687 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 649 . . . . 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 310 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1110albii 1620 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3493 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3800 . 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 369   A.wal 1377    = wceq 1379    e. wcel 1767    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by:  disjdif  3899  inf3lem3  8043  ssfin4  8686  isnrm2  19625  1stccnp  19729  llycmpkgen2  19786  ufileu  20155  fclscf  20261  flimfnfcls  20264  inindif  27088  opnbnd  29720  diophrw  30296  setindtr  30570
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