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Theorem inpws2 25043
Description: An intersection with a member of a powerset belongs to this powerset. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
inpws2  |-  ( B  e.  ~P C  -> 
( A  i^i  B
)  e.  ~P C
)

Proof of Theorem inpws2
StepHypRef Expression
1 incom 3361 . 2  |-  ( A  i^i  B )  =  ( B  i^i  A
)
2 inpws1 25042 . 2  |-  ( B  e.  ~P C  -> 
( B  i^i  A
)  e.  ~P C
)
31, 2syl5eqel 2367 1  |-  ( B  e.  ~P C  -> 
( A  i^i  B
)  e.  ~P C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151   ~Pcpw 3625
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  ax-sep 4141
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-in 3159  df-ss 3166  df-pw 3627
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