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Theorem undifabs 3531
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 3429 . 2  |-  ( A  u.  ( A  \  B ) )  =  ( ( A  u.  A )  \  ( B  \  A ) )
2 unidm 3318 . . 3  |-  ( A  u.  A )  =  A
32difeq1i 3290 . 2  |-  ( ( A  u.  A ) 
\  ( B  \  A ) )  =  ( A  \  ( B  \  A ) )
4 difdif 3302 . 2  |-  ( A 
\  ( B  \  A ) )  =  A
51, 3, 43eqtri 2307 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    u. cun 3150
This theorem is referenced by:  dfif5  3577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157
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