MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difdif2 Structured version   Unicode version

Theorem difdif2 3619
Description: Set difference with a set difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
Assertion
Ref Expression
difdif2  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )

Proof of Theorem difdif2
StepHypRef Expression
1 difindi 3616 . 2  |-  ( A 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
2 invdif 3603 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
32eqcomi 2447 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
43difeq2i 3483 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
5 dfin2 3598 . . 3  |-  ( A  i^i  C )  =  ( A  \  ( _V  \  C ) )
65uneq2i 3519 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
71, 4, 63eqtr4i 2473 1  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   _Vcvv 2984    \ cdif 3337    u. cun 3338    i^i cin 3339
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 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347
This theorem is referenced by:  restmetu  20174  mblfinlem3  28442  mblfinlem4  28443
  Copyright terms: Public domain W3C validator