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Theorem difsnid 3887
Description: If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)
Assertion
Ref Expression
difsnid  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )

Proof of Theorem difsnid
StepHypRef Expression
1 uncom 3434 . 2  |-  ( ( A  \  { B } )  u.  { B } )  =  ( { B }  u.  ( A  \  { B } ) )
2 snssi 3885 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
3 undif 3651 . . 3  |-  ( { B }  C_  A  <->  ( { B }  u.  ( A  \  { B } ) )  =  A )
42, 3sylib 189 . 2  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
51, 4syl5eq 2431 1  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    \ cdif 3260    u. cun 3261    C_ wss 3263   {csn 3757
This theorem is referenced by:  fnsnsplit  5869  fsnunf2  5871  difsnen  7126  phplem2  7223  pssnn  7263  dif1enOLD  7276  dif1en  7277  frfi  7288  xpfi  7314  dif1card  7825  hashfun  11627  mreexexlem4d  13799  tdeglem4  19850  dfconngra1  21506  enfixsn  26926  islindf4  26977  hdmap14lem4a  31989  hdmap14lem13  31998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-sn 3763
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