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Theorem moec 25150
Description: Moving an element  B out from the intersection of a class  A. (Contributed by FL, 29-Nov-2007.)
Assertion
Ref Expression
moec  |-  ( B  e.  A  ->  |^| A  =  ( B  i^i  |^| ( A  \  { B } ) ) )

Proof of Theorem moec
StepHypRef Expression
1 difsnid 3777 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
21eqcomd 2301 . . 3  |-  ( B  e.  A  ->  A  =  ( ( A 
\  { B }
)  u.  { B } ) )
32inteqd 3883 . 2  |-  ( B  e.  A  ->  |^| A  =  |^| ( ( A 
\  { B }
)  u.  { B } ) )
4 intun 3910 . . 3  |-  |^| (
( A  \  { B } )  u.  { B } )  =  (
|^| ( A  \  { B } )  i^i  |^| { B } )
5 intsng 3913 . . . . 5  |-  ( B  e.  A  ->  |^| { B }  =  B )
65ineq2d 3383 . . . 4  |-  ( B  e.  A  ->  ( |^| ( A  \  { B } )  i^i  |^| { B } )  =  ( |^| ( A 
\  { B }
)  i^i  B )
)
7 incom 3374 . . . 4  |-  ( |^| ( A  \  { B } )  i^i  B
)  =  ( B  i^i  |^| ( A  \  { B } ) )
86, 7syl6eq 2344 . . 3  |-  ( B  e.  A  ->  ( |^| ( A  \  { B } )  i^i  |^| { B } )  =  ( B  i^i  |^| ( A  \  { B } ) ) )
94, 8syl5eq 2340 . 2  |-  ( B  e.  A  ->  |^| (
( A  \  { B } )  u.  { B } )  =  ( B  i^i  |^| ( A  \  { B }
) ) )
103, 9eqtrd 2328 1  |-  ( B  e.  A  ->  |^| A  =  ( B  i^i  |^| ( A  \  { B } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164   {csn 3653   |^|cint 3878
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-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-int 3879
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