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Theorem difdif 3277
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif  |-  ( A 
\  ( B  \  A ) )  =  A

Proof of Theorem difdif
StepHypRef Expression
1 pm4.45im 547 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
2 iman 415 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
3 eldif 3137 . . . . 5  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
42, 3xchbinxr 304 . . . 4  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  x  e.  ( B 
\  A ) )
54anbi2i 678 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
61, 5bitr2i 243 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
76difeqri 3271 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3124
This theorem is referenced by:  dif0  3499  undifabs  3506
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-dif 3130
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