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Theorem difsnid 3936
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 3483 . 2  |-  ( ( A  \  { B } )  u.  { B } )  =  ( { B }  u.  ( A  \  { B } ) )
2 snssi 3934 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
3 undif 3700 . . 3  |-  ( { B }  C_  A  <->  ( { B }  u.  ( A  \  { B } ) )  =  A )
42, 3sylib 189 . 2  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
51, 4syl5eq 2479 1  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    C_ wss 3312   {csn 3806
This theorem is referenced by:  fnsnsplit  5922  fsnunf2  5924  difsnen  7182  phplem2  7279  pssnn  7319  dif1enOLD  7332  dif1en  7333  frfi  7344  xpfi  7370  dif1card  7884  hashfun  11692  mreexexlem4d  13864  tdeglem4  19975  dfconngra1  21650  enfixsn  27225  islindf4  27276  hdmap14lem4a  32609  hdmap14lem13  32618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812
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