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Theorem undifabs 3706
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 undif3 3603 . 2  |-  ( A  u.  ( A  \  B ) )  =  ( ( A  u.  A )  \  ( B  \  A ) )
2 unidm 3491 . . 3  |-  ( A  u.  A )  =  A
32difeq1i 3462 . 2  |-  ( ( A  u.  A ) 
\  ( B  \  A ) )  =  ( A  \  ( B  \  A ) )
4 difdif 3474 . 2  |-  ( A 
\  ( B  \  A ) )  =  A
51, 3, 43eqtri 2461 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3318    u. cun 3319
This theorem is referenced by:  dfif5  3752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326
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