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Theorem difin 3308
Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )

Proof of Theorem difin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-in2 604 . . . . . . . 8  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( (
x  e.  A  /\  x  e.  B )  -> F.  ) )
21expd 256 . . . . . . 7  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( x  e.  A  ->  ( x  e.  B  -> F.  ) ) )
3 dfnot 1349 . . . . . . 7  |-  ( -.  x  e.  B  <->  ( x  e.  B  -> F.  )
)
42, 3syl6ibr 161 . . . . . 6  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( x  e.  A  ->  -.  x  e.  B ) )
54com12 30 . . . . 5  |-  ( x  e.  A  ->  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  -.  x  e.  B ) )
65imdistani 441 . . . 4  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  ->  (
x  e.  A  /\  -.  x  e.  B
) )
7 simpr 109 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  B )
87con3i 621 . . . . 5  |-  ( -.  x  e.  B  ->  -.  ( x  e.  A  /\  x  e.  B
) )
98anim2i 339 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
106, 9impbii 125 . . 3  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
11 eldif 3075 . . . 4  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B
) ) )
12 elin 3254 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1312notbii 657 . . . . 5  |-  ( -.  x  e.  ( A  i^i  B )  <->  -.  (
x  e.  A  /\  x  e.  B )
)
1413anbi2i 452 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B ) ) )
1511, 14bitri 183 . . 3  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
16 eldif 3075 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1710, 15, 163bitr4i 211 . 2  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  x  e.  ( A  \  B ) )
1817eqriv 2134 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331   F. wfal 1336    e. wcel 1480    \ cdif 3063    i^i cin 3065
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072
This theorem is referenced by:  inssddif  3312  symdif1  3336  notrab  3348  disjdif2  3436  unfiin  6807
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