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Theorem difpr 3574
Description: Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
Assertion
Ref Expression
difpr  |-  ( A 
\  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } )

Proof of Theorem difpr
StepHypRef Expression
1 df-pr 3448 . . 3  |-  { B ,  C }  =  ( { B }  u.  { C } )
21difeq2i 3113 . 2  |-  ( A 
\  { B ,  C } )  =  ( A  \  ( { B }  u.  { C } ) )
3 difun1 3257 . 2  |-  ( A 
\  ( { B }  u.  { C } ) )  =  ( ( A  \  { B } )  \  { C } )
42, 3eqtri 2108 1  |-  ( A 
\  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } )
Colors of variables: wff set class
Syntax hints:    = wceq 1289    \ cdif 2994    u. cun 2995   {csn 3441   {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-pr 3448
This theorem is referenced by:  hashdifpr  10193
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