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Theorem pwss 3494
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 3054 . 2  |-  ( ~P A  C_  B  <->  A. x
( x  e.  ~P A  ->  x  e.  B
) )
2 df-pw 3480 . . . . 5  |-  ~P A  =  { x  |  x 
C_  A }
32abeq2i 2226 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
43imbi1i 237 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  B )  <-> 
( x  C_  A  ->  x  e.  B ) )
54albii 1429 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  B )  <->  A. x
( x  C_  A  ->  x  e.  B ) )
61, 5bitri 183 1  |-  ( ~P A  C_  B  <->  A. x
( x  C_  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312    e. wcel 1463    C_ wss 3039   ~Pcpw 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  axpweq  4063  setind2  4423
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