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Theorem undifabs 3365
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs  |-  ( A  u.  ( A  \  B ) )  =  A

Proof of Theorem undifabs
StepHypRef Expression
1 ssid 3047 . . 3  |-  A  C_  A
2 difss 3129 . . 3  |-  ( A 
\  B )  C_  A
31, 2unssi 3178 . 2  |-  ( A  u.  ( A  \  B ) )  C_  A
4 ssun1 3166 . 2  |-  A  C_  ( A  u.  ( A  \  B ) )
53, 4eqssi 3044 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1290    \ cdif 2999    u. cun 3000
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
This theorem is referenced by: (None)
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