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Theorem difun2 3533
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 3422 . 2  |-  ( ( A  u.  B ) 
\  B )  =  ( ( A  \  B )  u.  ( B  \  B ) )
2 difid 3522 . . 3  |-  ( B 
\  B )  =  (/)
32uneq2i 3326 . 2  |-  ( ( A  \  B )  u.  ( B  \  B ) )  =  ( ( A  \  B )  u.  (/) )
4 un0 3479 . 2  |-  ( ( A  \  B )  u.  (/) )  =  ( A  \  B )
51, 3, 43eqtri 2307 1  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    u. cun 3150   (/)c0 3455
This theorem is referenced by:  uneqdifeq  3542  orddif  4486  domunsncan  6962  elfiun  7183  hartogslem1  7257  cantnfp1lem3  7382  cda1dif  7802  infcda1  7819  ssxr  8892  dfn2  9978  incexclem  12295  mreexmrid  13545  lbsextlem4  15914  ufprim  17604  volun  18902  i1f1  19045  itgioo  19170  itgsplitioo  19192  plyeq0lem  19592  jensen  20283  difeq  23128  difprsn2  23191  measxun  23539  kelac2  27163  pwfi2f1o  27260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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