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Theorem difsnid 3777
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 3332 . 2  |-  ( ( A  \  { B } )  u.  { B } )  =  ( { B }  u.  ( A  \  { B } ) )
2 snssi 3775 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
3 undif 3547 . . 3  |-  ( { B }  C_  A  <->  ( { B }  u.  ( A  \  { B } ) )  =  A )
42, 3sylib 188 . 2  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
51, 4syl5eq 2340 1  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653
This theorem is referenced by:  fnsnsplit  5733  fsnunf2  5735  difsnen  6960  phplem2  7057  pssnn  7097  dif1enOLD  7106  dif1en  7107  frfi  7118  xpfi  7144  dif1card  7654  hashfun  11405  mreexexlem4d  13565  tdeglem4  19462  moec  25150  enfixsn  27360  islindf4  27411  hdmap14lem4a  32686  hdmap14lem13  32695
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
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