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Theorem undifabs 3321
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 3019 . . 3  |-  A  C_  A
2 difss 3099 . . 3  |-  ( A 
\  B )  C_  A
31, 2unssi 3148 . 2  |-  ( A  u.  ( A  \  B ) )  C_  A
4 ssun1 3136 . 2  |-  A  C_  ( A  u.  ( A  \  B ) )
53, 4eqssi 3016 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    \ cdif 2971    u. cun 2972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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