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Theorem inabs 3557
Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
Assertion
Ref Expression
inabs  |-  ( A  i^i  ( A  u.  B ) )  =  A

Proof of Theorem inabs
StepHypRef Expression
1 ssun1 3496 . 2  |-  A  C_  ( A  u.  B
)
2 df-ss 3320 . 2  |-  ( A 
C_  ( A  u.  B )  <->  ( A  i^i  ( A  u.  B
) )  =  A )
31, 2mpbi 201 1  |-  ( A  i^i  ( A  u.  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    u. cun 3304    i^i cin 3305    C_ wss 3306
This theorem is referenced by:  dfif5  3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-un 3311  df-in 3313  df-ss 3320
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