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Theorem undifabs 3812
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 3672 . 2  |-  ( A  u.  ( A  \  B ) )  =  ( ( A  u.  A )  \  ( B  \  A ) )
2 unidm 3547 . . 3  |-  ( A  u.  A )  =  A
32difeq1i 3517 . 2  |-  ( ( A  u.  A ) 
\  ( B  \  A ) )  =  ( A  \  ( B  \  A ) )
4 difdif 3529 . 2  |-  ( A 
\  ( B  \  A ) )  =  A
51, 3, 43eqtri 2449 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    \ cdif 3371    u. cun 3372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379
This theorem is referenced by:  dfif5  3865  indifundif  28090
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