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Theorem undifabs 3490
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 3167 . . 3  |-  A  C_  A
2 difss 3253 . . 3  |-  ( A 
\  B )  C_  A
31, 2unssi 3302 . 2  |-  ( A  u.  ( A  \  B ) )  C_  A
4 ssun1 3290 . 2  |-  A  C_  ( A  u.  ( A  \  B ) )
53, 4eqssi 3163 1  |-  ( A  u.  ( A  \  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1348    \ cdif 3118    u. cun 3119
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by:  exmid1stab  13993
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