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Theorem difin 3689
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 pm4.61 426 . . 3  |-  ( -.  ( x  e.  A  ->  x  e.  B )  <-> 
( x  e.  A  /\  -.  x  e.  B
) )
2 anclb 547 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  -> 
( x  e.  A  /\  x  e.  B
) ) )
3 elin 3628 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
43imbi2i 312 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  ( x  e.  A  ->  ( x  e.  A  /\  x  e.  B ) ) )
5 iman 424 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  -.  (
x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
62, 4, 53bitr2i 275 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  -.  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
76con2bii 332 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <->  -.  ( x  e.  A  ->  x  e.  B ) )
8 eldif 3426 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
91, 7, 83bitr4i 279 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
x  e.  ( A 
\  B ) )
109difeqri 3565 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    \ cdif 3413    i^i cin 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-dif 3419  df-in 3423
This theorem is referenced by:  dfin4  3692  indif  3694  dfsymdif3  3717  symdif1OLD  3718  notrab  3729  dfsdom2  7680  hashdif  12527  isercolllem3  13640  iuncld  19840  llycmpkgen2  20345  1stckgen  20349  ptbasfi  20376  txkgen  20447  cmmbl  22239  disjdifprg2  27881  ldgenpisyslem1  28624  onint1  30694  bj-disjdif  31089  nzprmdif  36085  dvmptfprodlem  37122
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