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Theorem undifabs 3439
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 3117 . . 3  |-  A  C_  A
2 difss 3202 . . 3  |-  ( A 
\  B )  C_  A
31, 2unssi 3251 . 2  |-  ( A  u.  ( A  \  B ) )  C_  A
4 ssun1 3239 . 2  |-  A  C_  ( A  u.  ( A  \  B ) )
53, 4eqssi 3113 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1331    \ cdif 3068    u. cun 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084
This theorem is referenced by:  exmid1stab  13225
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