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

Theorem unundir 3350
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundir  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)

Proof of Theorem unundir
StepHypRef Expression
1 unidm 3331 . . 3  |-  ( C  u.  C )  =  C
21uneq2i 3339 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  C ) )  =  ( ( A  u.  B )  u.  C
)
3 un4 3348 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  C ) )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)
42, 3eqtr3i 2318 1  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
  Copyright terms: Public domain W3C validator