Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inpws1 Unicode version

Theorem inpws1 25042
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 4157 . 2  |-  ( A  e.  ~P C  -> 
( A  i^i  B
)  e.  _V )
2 elpwi 3633 . . . 4  |-  ( A  e.  ~P C  ->  A  C_  C )
3 ssinss1 3397 . . . 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 3632 . . 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 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  inpws2  25043
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
  Copyright terms: Public domain W3C validator