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Theorem difun2 3838
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 3687 . 2  |-  ( ( A  u.  B ) 
\  B )  =  ( ( A  \  B )  u.  ( B  \  B ) )
2 difid 3747 . . 3  |-  ( B 
\  B )  =  (/)
32uneq2i 3576 . 2  |-  ( ( A  \  B )  u.  ( B  \  B ) )  =  ( ( A  \  B )  u.  (/) )
4 un0 3762 . 2  |-  ( ( A  \  B )  u.  (/) )  =  ( A  \  B )
51, 3, 43eqtri 2497 1  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    \ cdif 3387    u. cun 3388   (/)c0 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723
This theorem is referenced by:  uneqdifeq  3847  difprsn1  4099  orddif  5523  domunsncan  7690  elfiun  7962  hartogslem1  8075  cantnfp1lem3  8203  cda1dif  8624  infcda1  8641  ssxr  9721  dfn2  10906  incexclem  13971  mreexmrid  15627  lbsextlem4  18462  ufprim  21002  volun  22577  i1f1  22727  itgioo  22852  itgsplitioo  22874  plyeq0lem  23243  jensen  23993  difeq  28230  fzdif2  28444  measun  29107  carsgclctunlem1  29222  carsggect  29223  elmrsubrn  30230  mrsubvrs  30232  finixpnum  31994  poimirlem2  32006  poimirlem4  32008  poimirlem6  32010  poimirlem7  32011  poimirlem8  32012  poimirlem11  32015  poimirlem12  32016  poimirlem13  32017  poimirlem14  32018  poimirlem16  32020  poimirlem18  32022  poimirlem19  32023  poimirlem21  32025  poimirlem23  32027  poimirlem27  32031  poimirlem30  32034  asindmre  32091  kelac2  35994  pwfi2f1o  36025  iccdifioo  37712  iccdifprioo  37713  hoiprodp1  38528
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