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Theorem difsnss 3533
Description: If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6139. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
difsnss  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  C_  A
)

Proof of Theorem difsnss
StepHypRef Expression
1 uncom 3117 . 2  |-  ( ( A  \  { B } )  u.  { B } )  =  ( { B }  u.  ( A  \  { B } ) )
2 snssi 3531 . . 3  |-  ( B  e.  A  ->  { B }  C_  A )
3 undifss 3324 . . 3  |-  ( { B }  C_  A  <->  ( { B }  u.  ( A  \  { B } ) )  C_  A )
42, 3sylib 120 . 2  |-  ( B  e.  A  ->  ( { B }  u.  ( A  \  { B }
) )  C_  A
)
51, 4syl5eqss 3044 1  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    \ cdif 2971    u. cun 2972    C_ wss 2974   {csn 3400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406
This theorem is referenced by:  nndifsnid  6139  fidifsnid  6396
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