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Theorem difin 3387
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 616 . . . . . . . 8  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( (
x  e.  A  /\  x  e.  B )  -> F.  ) )
21expd 258 . . . . . . 7  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  ->  ( x  e.  A  ->  ( x  e.  B  -> F.  ) ) )
3 dfnot 1382 . . . . . . 7  |-  ( -.  x  e.  B  <->  ( x  e.  B  -> F.  )
)
42, 3imbitrrdi 162 . . . . . 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 445 . . . 4  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  ->  (
x  e.  A  /\  -.  x  e.  B
) )
7 simpr 110 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  B )
87con3i 633 . . . . 5  |-  ( -.  x  e.  B  ->  -.  ( x  e.  A  /\  x  e.  B
) )
98anim2i 342 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
106, 9impbii 126 . . 3  |-  ( ( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B
) )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
11 eldif 3153 . . . 4  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B
) ) )
12 elin 3333 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
1312notbii 669 . . . . 5  |-  ( -.  x  e.  ( A  i^i  B )  <->  -.  (
x  e.  A  /\  x  e.  B )
)
1413anbi2i 457 . . . 4  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
( x  e.  A  /\  -.  ( x  e.  A  /\  x  e.  B ) ) )
1511, 14bitri 184 . . 3  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  ( x  e.  A  /\  -.  (
x  e.  A  /\  x  e.  B )
) )
16 eldif 3153 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1710, 15, 163bitr4i 212 . 2  |-  ( x  e.  ( A  \ 
( A  i^i  B
) )  <->  x  e.  ( A  \  B ) )
1817eqriv 2186 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1364   F. wfal 1369    e. wcel 2160    \ cdif 3141    i^i cin 3143
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150
This theorem is referenced by:  inssddif  3391  symdif1  3415  notrab  3427  disjdif2  3516  unfiin  6955  bj-charfundcALT  15039
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