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Theorem inssdif0 3863
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 3650 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 327 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 iman 426 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  <->  -.  (
( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C ) )
42, 3bitri 253 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3447 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 699 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3650 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 654 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 282 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9xchbinx 312 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1110albii 1688 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3454 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3778 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133bitr4i 281 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 188    /\ wa 371   A.wal 1436    = wceq 1438    e. wcel 1869    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763
This theorem is referenced by:  disjdif  3868  inf3lem3  8139  ssfin4  8742  isnrm2  20366  1stccnp  20469  llycmpkgen2  20557  ufileu  20926  fclscf  21032  flimfnfcls  21035  inindif  28143  opnbnd  30980  diophrw  35564  setindtr  35843
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