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Theorem inssdif0im 3347
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
inssdif0im  |-  ( ( A  i^i  B ) 
C_  C  ->  ( A  i^i  ( B  \  C ) )  =  (/) )

Proof of Theorem inssdif0im
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3181 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 236 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 imanim 823 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
42, 3sylbi 119 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3006 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 445 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3181 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 393 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 211 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9sylnib 636 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  x  e.  ( A  i^i  ( B 
\  C ) ) )
1110alimi 1389 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  ->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3012 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3299 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133imtr4i 199 1  |-  ( ( A  i^i  B ) 
C_  C  ->  ( A  i^i  ( B  \  C ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438    \ cdif 2994    i^i cin 2996    C_ wss 2997   (/)c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285
This theorem is referenced by:  disjdif  3352
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