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Theorem inundif 3545
Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
inundif  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A

Proof of Theorem inundif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3371 . . . 4  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2 eldif 3175 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
31, 2orbi12i 507 . . 3  |-  ( ( x  e.  ( A  i^i  B )  \/  x  e.  ( A 
\  B ) )  <-> 
( ( x  e.  A  /\  x  e.  B )  \/  (
x  e.  A  /\  -.  x  e.  B
) ) )
4 pm4.42 926 . . 3  |-  ( x  e.  A  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  -.  x  e.  B
) ) )
53, 4bitr4i 243 . 2  |-  ( ( x  e.  ( A  i^i  B )  \/  x  e.  ( A 
\  B ) )  <-> 
x  e.  A )
65uneqri 3330 1  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164
This theorem is referenced by:  resasplit  5427  fresaun  5428  fresaunres2  5429  ixpfi2  7170  hashun3  11382  prmreclem2  12980  sylow2a  14946  ablfac1eu  15324  basdif0  16707  cmpfi  17151  ptbasfi  17292  ptcnplem  17331  fin1aufil  17643  ismbl2  18902  volinun  18919  voliunlem2  18924  mbfeqalem  19013  itg2cnlem2  19133  dvres2lem  19276  partfun  23255  measxun  23554  probdif  23638  islimrs4  25685  mvdco  27491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170  df-in 3172
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