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Theorem difin 3234
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 580 . . . . . . . 8  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( (
x  e.  A  /\  x  e.  B )  -> F.  ) )
21expd 254 . . . . . . 7  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( x  e.  A  ->  ( x  e.  B  -> F.  ) ) )
3 dfnot 1307 . . . . . . 7  |-  ( -.  x  e.  B  <->  ( x  e.  B  -> F.  )
)
42, 3syl6ibr 160 . . . . . 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 434 . . . 4  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  ->  (
x  e.  A  /\  -.  x  e.  B
) )
7 simpr 108 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  B )
87con3i 597 . . . . 5  |-  ( -.  x  e.  B  ->  -.  ( x  e.  A  /\  x  e.  B
) )
98anim2i 334 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
106, 9impbii 124 . . 3  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
11 eldif 3006 . . . 4  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B
) ) )
12 elin 3181 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1312notbii 629 . . . . 5  |-  ( -.  x  e.  ( A  i^i  B )  <->  -.  (
x  e.  A  /\  x  e.  B )
)
1413anbi2i 445 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B ) ) )
1511, 14bitri 182 . . 3  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
16 eldif 3006 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1710, 15, 163bitr4i 210 . 2  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  x  e.  ( A  \  B ) )
1817eqriv 2085 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289   F. wfal 1294    e. wcel 1438    \ cdif 2994    i^i cin 2996
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-fal 1295  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
This theorem is referenced by:  inssddif  3238  symdif1  3262  notrab  3274  unfiin  6616
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