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Theorem inpws1 25145
Description: An intersection with a member of a powerset belongs to this powerset. (Contributed by FL, 25-Sep-2007.)
Assertion
Ref Expression
inpws1  |-  ( A  e.  ~P C  -> 
( A  i^i  B
)  e.  ~P C
)

Proof of Theorem inpws1
StepHypRef Expression
1 inex1g 4173 . 2  |-  ( A  e.  ~P C  -> 
( A  i^i  B
)  e.  _V )
2 elpwi 3646 . . . 4  |-  ( A  e.  ~P C  ->  A  C_  C )
3 ssinss1 3410 . . . 4  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
42, 3syl 15 . . 3  |-  ( A  e.  ~P C  -> 
( A  i^i  B
)  C_  C )
5 elpwg 3645 . . 3  |-  ( ( A  i^i  B )  e.  _V  ->  (
( A  i^i  B
)  e.  ~P C  <->  ( A  i^i  B ) 
C_  C ) )
64, 5syl5ibr 212 . 2  |-  ( ( A  i^i  B )  e.  _V  ->  ( A  e.  ~P C  ->  ( A  i^i  B
)  e.  ~P C
) )
71, 6mpcom 32 1  |-  ( A  e.  ~P C  -> 
( A  i^i  B
)  e.  ~P C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  inpws2  25146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
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