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Theorem undifabs 3544
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 3442 . 2  |-  ( A  u.  ( A  \  B ) )  =  ( ( A  u.  A )  \  ( B  \  A ) )
2 unidm 3331 . . 3  |-  ( A  u.  A )  =  A
32difeq1i 3303 . 2  |-  ( ( A  u.  A ) 
\  ( B  \  A ) )  =  ( A  \  ( B  \  A ) )
4 difdif 3315 . 2  |-  ( A 
\  ( B  \  A ) )  =  A
51, 3, 43eqtri 2320 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    u. cun 3163
This theorem is referenced by:  dfif5  3590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170
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