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Theorem inabs 3400
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 3338 . 2  |-  A  C_  ( A  u.  B
)
2 df-ss 3166 . 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 1623    u. cun 3150    i^i cin 3151    C_ wss 3152
This theorem is referenced by:  dfif5  3577  inabs2  25138  hdrmp  25706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166
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