MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difpr Structured version   Unicode version

Theorem difpr 4110
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 3974 . . 3  |-  { B ,  C }  =  ( { B }  u.  { C } )
21difeq2i 3557 . 2  |-  ( A 
\  { B ,  C } )  =  ( A  \  ( { B }  u.  { C } ) )
3 difun1 3709 . 2  |-  ( A 
\  ( { B }  u.  { C } ) )  =  ( ( A  \  { B } )  \  { C } )
42, 3eqtri 2431 1  |-  ( A 
\  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    \ cdif 3410    u. cun 3411   {csn 3971   {cpr 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-pr 3974
This theorem is referenced by:  nbgrassvwo2  24842
  Copyright terms: Public domain W3C validator