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Theorem difun2 3758
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 3603 . 2  |-  ( ( A  u.  B ) 
\  B )  =  ( ( A  \  B )  u.  ( B  \  B ) )
2 difid 3747 . . 3  |-  ( B 
\  B )  =  (/)
32uneq2i 3507 . 2  |-  ( ( A  \  B )  u.  ( B  \  B ) )  =  ( ( A  \  B )  u.  (/) )
4 un0 3662 . 2  |-  ( ( A  \  B )  u.  (/) )  =  ( A  \  B )
51, 3, 43eqtri 2467 1  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    \ cdif 3325    u. cun 3326   (/)c0 3637
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 2568  df-ne 2608  df-ral 2720  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638
This theorem is referenced by:  uneqdifeq  3767  difprsn1  4010  orddif  4812  domunsncan  7411  elfiun  7680  hartogslem1  7756  cantnfp1lem3  7888  cantnfp1lem3OLD  7914  cda1dif  8345  infcda1  8362  ssxr  9444  dfn2  10592  incexclem  13299  mreexmrid  14581  lbsextlem4  17242  ufprim  19482  volun  21026  i1f1  21168  itgioo  21293  itgsplitioo  21315  plyeq0lem  21678  jensen  22382  difeq  25899  measun  26625  finixpnum  28414  asindmre  28479  kelac2  29418  pwfi2f1o  29451
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