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Theorem moec 24459
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 3761 . . . 4  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
21eqcomd 2288 . . 3  |-  ( B  e.  A  ->  A  =  ( ( A 
\  { B }
)  u.  { B } ) )
32inteqd 3867 . 2  |-  ( B  e.  A  ->  |^| A  =  |^| ( ( A 
\  { B }
)  u.  { B } ) )
4 intun 3894 . . 3  |-  |^| (
( A  \  { B } )  u.  { B } )  =  (
|^| ( A  \  { B } )  i^i  |^| { B } )
5 intsng 3897 . . . . 5  |-  ( B  e.  A  ->  |^| { B }  =  B )
65ineq2d 3370 . . . 4  |-  ( B  e.  A  ->  ( |^| ( A  \  { B } )  i^i  |^| { B } )  =  ( |^| ( A 
\  { B }
)  i^i  B )
)
7 incom 3361 . . . 4  |-  ( |^| ( A  \  { B } )  i^i  B
)  =  ( B  i^i  |^| ( A  \  { B } ) )
86, 7syl6eq 2331 . . 3  |-  ( B  e.  A  ->  ( |^| ( A  \  { B } )  i^i  |^| { B } )  =  ( B  i^i  |^| ( A  \  { B } ) ) )
94, 8syl5eq 2327 . 2  |-  ( B  e.  A  ->  |^| (
( A  \  { B } )  u.  { B } )  =  ( B  i^i  |^| ( A  \  { B }
) ) )
103, 9eqtrd 2315 1  |-  ( B  e.  A  ->  |^| A  =  ( B  i^i  |^| ( A  \  { B } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151   {csn 3640   |^|cint 3862
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-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-int 3863
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