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Theorem difpr 3553
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 3429 . . 3  |-  { B ,  C }  =  ( { B }  u.  { C } )
21difeq2i 3099 . 2  |-  ( A 
\  { B ,  C } )  =  ( A  \  ( { B }  u.  { C } ) )
3 difun1 3242 . 2  |-  ( A 
\  ( { B }  u.  { C } ) )  =  ( ( A  \  { B } )  \  { C } )
42, 3eqtri 2103 1  |-  ( A 
\  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    \ cdif 2981    u. cun 2982   {csn 3422   {cpr 3423
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-pr 3429
This theorem is referenced by:  hashdifpr  10063
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