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Theorem difin 3406
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 pm4.61 415 . . 3  |-  ( -.  ( x  e.  A  ->  x  e.  B )  <-> 
( x  e.  A  /\  -.  x  e.  B
) )
2 anclb 530 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  -> 
( x  e.  A  /\  x  e.  B
) ) )
3 elin 3358 . . . . . 6  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
43imbi2i 303 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  ( x  e.  A  ->  ( x  e.  A  /\  x  e.  B ) ) )
5 iman 413 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  ( A  i^i  B ) )  <->  -.  (
x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
62, 4, 53bitr2i 264 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  -.  ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) ) )
76con2bii 322 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <->  -.  ( x  e.  A  ->  x  e.  B ) )
8 eldif 3162 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
91, 7, 83bitr4i 268 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( A  i^i  B ) )  <-> 
x  e.  ( A 
\  B ) )
109difeqri 3296 1  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151
This theorem is referenced by:  dfin4  3409  indif  3411  symdif1  3433  notrab  3445  dfsdom2  6984  hashdif  11375  isercolllem3  12140  iuncld  16782  llycmpkgen2  17245  1stckgen  17249  ptbasfi  17276  txkgen  17346  cmmbl  18892  disjdifprg2  23353  onint1  24888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159
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