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Theorem inssdif0im 3490
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 3318 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21imbi1i 238 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C ) )
3 imanim 688 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
42, 3sylbi 121 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  ( ( x  e.  A  /\  x  e.  B )  /\  -.  x  e.  C )
)
5 eldif 3138 . . . . . 6  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
65anbi2i 457 . . . . 5  |-  ( ( x  e.  A  /\  x  e.  ( B  \  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
7 elin 3318 . . . . 5  |-  ( x  e.  ( A  i^i  ( B  \  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  \  C
) ) )
8 anass 401 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  -.  x  e.  C
) ) )
96, 7, 83bitr4ri 213 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  -.  x  e.  C )  <->  x  e.  ( A  i^i  ( B  \  C ) ) )
104, 9sylnib 676 . . 3  |-  ( ( x  e.  ( A  i^i  B )  ->  x  e.  C )  ->  -.  x  e.  ( A  i^i  ( B 
\  C ) ) )
1110alimi 1455 . 2  |-  ( A. x ( x  e.  ( A  i^i  B
)  ->  x  e.  C )  ->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
12 dfss2 3144 . 2  |-  ( ( A  i^i  B ) 
C_  C  <->  A. x
( x  e.  ( A  i^i  B )  ->  x  e.  C
) )
13 eq0 3441 . 2  |-  ( ( A  i^i  ( B 
\  C ) )  =  (/)  <->  A. x  -.  x  e.  ( A  i^i  ( B  \  C ) ) )
1411, 12, 133imtr4i 201 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 104   A.wal 1351    = wceq 1353    e. wcel 2148    \ cdif 3126    i^i cin 3128    C_ wss 3129   (/)c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  disjdif  3495
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