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Theorem difun2 3368
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 3255 . 2  |-  ( ( A  u.  B ) 
\  B )  =  ( ( A  \  B )  u.  ( B  \  B ) )
2 difid 3357 . . 3  |-  ( B 
\  B )  =  (/)
32uneq2i 3154 . 2  |-  ( ( A  \  B )  u.  ( B  \  B ) )  =  ( ( A  \  B )  u.  (/) )
4 un0 3322 . 2  |-  ( ( A  \  B )  u.  (/) )  =  ( A  \  B )
51, 3, 43eqtri 2113 1  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1290    \ cdif 2999    u. cun 3000   (/)c0 3289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290
This theorem is referenced by:  uneqdifeqim  3374  difprsn1  3584  orddif  4378  fisseneq  6698  dfn2  8749
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