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Theorem inabs 3488
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 3426 . 2  |-  A  C_  ( A  u.  B
)
2 df-ss 3252 . 2  |-  ( A 
C_  ( A  u.  B )  <->  ( A  i^i  ( A  u.  B
) )  =  A )
31, 2mpbi 199 1  |-  ( A  i^i  ( A  u.  B ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1647    u. cun 3236    i^i cin 3237    C_ wss 3238
This theorem is referenced by:  dfif5  3666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-un 3243  df-in 3245  df-ss 3252
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