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Theorem inssdif0im 3398
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 3227 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 237 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 imanim 660 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
42, 3sylbi 120 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3048 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 450 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3227 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 396 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 212 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9sylnib 648 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  x  e.  ( A  i^i  ( B 
\  C ) ) )
1110alimi 1414 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  ->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3054 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3349 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133imtr4i 200 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 103   A.wal 1312    = wceq 1314    e. wcel 1463    \ cdif 3036    i^i cin 3038    C_ wss 3039   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by:  disjdif  3403
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