MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwss Unicode version

Theorem pwss 3639
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 3169 . 2  |-  ( ~P A  C_  B  <->  A. x
( x  e.  ~P A  ->  x  e.  B
) )
2 df-pw 3627 . . . . 5  |-  ~P A  =  { x  |  x 
C_  A }
32abeq2i 2390 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
43imbi1i 315 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  B )  <-> 
( x  C_  A  ->  x  e.  B ) )
54albii 1553 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  B )  <->  A. x
( x  C_  A  ->  x  e.  B ) )
61, 5bitri 240 1  |-  ( ~P A  C_  B  <->  A. x
( x  C_  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  axpweq  4187  setind2  7420  axgroth5  8446  grothpw  8448  axgroth6  8450
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
This theorem depends on definitions:  df-bi 177  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-in 3159  df-ss 3166  df-pw 3627
  Copyright terms: Public domain W3C validator