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Theorem difsnid 4014
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 3495 . 2  |-  ( ( A  \  { B } )  u.  { B } )  =  ( { B }  u.  ( A  \  { B } ) )
2 snssi 4012 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
3 undif 3754 . . 3  |-  ( { B }  C_  A  <->  ( { B }  u.  ( A  \  { B } ) )  =  A )
42, 3sylib 196 . 2  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  =  A )
51, 4syl5eq 2482 1  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    \ cdif 3320    u. cun 3321    C_ wss 3323   {csn 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-sn 3873
This theorem is referenced by:  fnsnsplit  5910  fsnunf2  5912  difsnexi  6379  difsnen  7385  enfixsn  7412  phplem2  7483  pssnn  7523  dif1enOLD  7536  dif1en  7537  frfi  7549  xpfi  7575  dif1card  8169  hashfun  12191  mreexexlem4d  14577  symgextf1  15917  symgextfo  15918  symgfixf1  15934  gsummgp0  16687  islindf4  18242  gsummatr01  18440  tdeglem4  21504  dfconngra1  23508  mgpsumunsn  30710  gsumdifsnd  30713  gsumdifsndf  30716  hdmap14lem4a  35359  hdmap14lem13  35368
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