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Theorem difin 3735
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 3687 . . . . . 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 273 . . . 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 3486 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
91, 7, 83bitr4i 277 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
x  e.  ( A 
\  B ) )
109difeqri 3624 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    i^i cin 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-in 3483
This theorem is referenced by:  dfin4  3738  indif  3740  symdif1  3763  notrab  3775  dfsdom2  7637  hashdif  12437  isercolllem3  13448  iuncld  19312  llycmpkgen2  19786  1stckgen  19790  ptbasfi  19817  txkgen  19888  cmmbl  21680  disjdifprg2  27110  onint1  29491  nzprmdif  30824  bj-disjdif  33592
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