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Theorem difin 3407
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 )
Dummy variable  x is distinct from all other variables.

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 3359 . . . . . 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 3163 . . 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 3297 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    \ cdif 3150    i^i cin 3152
This theorem is referenced by:  dfin4  3410  indif  3412  symdif1  3434  notrab  3446  dfsdom2  6979  hashdif  11369  isercolllem3  12134  iuncld  16776  llycmpkgen2  17239  1stckgen  17243  ptbasfi  17270  txkgen  17340  cmmbl  18886  onint1  24295
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-dif 3156  df-in 3160
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