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Theorem difin 3313
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
StepHypRef Expression
1 pm4.61 417 . . 3  |-  ( -.  ( x  e.  A  ->  x  e.  B )  <-> 
( x  e.  A  /\  -.  x  e.  B
) )
2 anclb 532 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  -> 
( x  e.  A  /\  x  e.  B
) ) )
3 elin 3266 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
43imbi2i 305 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  ( x  e.  A  ->  ( x  e.  A  /\  x  e.  B ) ) )
5 iman 415 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  -.  (
x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
62, 4, 53bitr2i 266 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  -.  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
76con2bii 324 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <->  -.  ( x  e.  A  ->  x  e.  B ) )
8 eldif 3088 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
91, 7, 83bitr4i 270 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
x  e.  ( A 
\  B ) )
109difeqri 3213 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3075    i^i cin 3077
This theorem is referenced by:  dfin4  3316  indif  3318  symdif1  3340  notrab  3352  dfsdom2  6869  hashdif  11252  isercolllem3  12017  iuncld  16614  llycmpkgen2  17077  1stckgen  17081  ptbasfi  17108  txkgen  17178  cmmbl  18724  onint1  24062
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-in 3085
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