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

Theorem difdif2 3762
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 3759 . 2  |-  ( A 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
2 invdif 3746 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
32eqcomi 2470 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
43difeq2i 3615 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
5 dfin2 3741 . . 3  |-  ( A  i^i  C )  =  ( A  \  ( _V  \  C ) )
65uneq2i 3651 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
71, 4, 63eqtr4i 2496 1  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478
This theorem is referenced by:  restmetu  21215  difelcarsg  28440  mblfinlem3  30215  mblfinlem4  30216
  Copyright terms: Public domain W3C validator