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Theorem pwss 3440
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 3012 . 2  |-  ( ~P A  C_  B  <->  A. x
( x  e.  ~P A  ->  x  e.  B
) )
2 df-pw 3427 . . . . 5  |-  ~P A  =  { x  |  x 
C_  A }
32abeq2i 2198 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
43imbi1i 236 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  B )  <-> 
( x  C_  A  ->  x  e.  B ) )
54albii 1404 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  B )  <->  A. x
( x  C_  A  ->  x  e.  B ) )
61, 5bitri 182 1  |-  ( ~P A  C_  B  <->  A. x
( x  C_  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    e. wcel 1438    C_ wss 2997   ~Pcpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  axpweq  3998  setind2  4346
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