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Theorem difun2 3546
Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
difun2  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)

Proof of Theorem difun2
StepHypRef Expression
1 difundir 3435 . 2  |-  ( ( A  u.  B ) 
\  B )  =  ( ( A  \  B )  u.  ( B  \  B ) )
2 difid 3535 . . 3  |-  ( B 
\  B )  =  (/)
32uneq2i 3339 . 2  |-  ( ( A  \  B )  u.  ( B  \  B ) )  =  ( ( A  \  B )  u.  (/) )
4 un0 3492 . 2  |-  ( ( A  \  B )  u.  (/) )  =  ( A  \  B )
51, 3, 43eqtri 2320 1  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    u. cun 3163   (/)c0 3468
This theorem is referenced by:  uneqdifeq  3555  difprsn1  3770  orddif  4502  domunsncan  6978  elfiun  7199  hartogslem1  7273  cantnfp1lem3  7398  cda1dif  7818  infcda1  7835  ssxr  8908  dfn2  9994  incexclem  12311  mreexmrid  13561  lbsextlem4  15930  ufprim  17620  volun  18918  i1f1  19061  itgioo  19186  itgsplitioo  19208  plyeq0lem  19608  jensen  20299  difeq  23144  measxun  23554  kelac2  27266  pwfi2f1o  27363
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-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469
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