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Theorem pwss 3805
Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
Assertion
Ref Expression
pwss  |-  ( ~P A  C_  B  <->  A. x
( x  C_  A  ->  x  e.  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem pwss
StepHypRef Expression
1 dfss2 3329 . 2  |-  ( ~P A  C_  B  <->  A. x
( x  e.  ~P A  ->  x  e.  B
) )
2 df-pw 3793 . . . . 5  |-  ~P A  =  { x  |  x 
C_  A }
32abeq2i 2542 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
43imbi1i 316 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  B )  <-> 
( x  C_  A  ->  x  e.  B ) )
54albii 1575 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  B )  <->  A. x
( x  C_  A  ->  x  e.  B ) )
61, 5bitri 241 1  |-  ( ~P A  C_  B  <->  A. x
( x  C_  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725    C_ wss 3312   ~Pcpw 3791
This theorem is referenced by:  axpweq  4368  setind2  7666  axgroth5  8691  grothpw  8693  axgroth6  8695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-pw 3793
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