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

Theorem ssextss 4227
Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssextss  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
Distinct variable groups:    x, A    x, B

Proof of Theorem ssextss
StepHypRef Expression
1 sspwb 4223 . 2  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 dfss2 3169 . 2  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
3 vex 2791 . . . . 5  |-  x  e. 
_V
43elpw 3631 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3631 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5imbi12i 316 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  ~P B
)  <->  ( x  C_  A  ->  x  C_  B
) )
76albii 1553 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  <->  A. x
( x  C_  A  ->  x  C_  B )
)
81, 2, 73bitri 262 1  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  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:  ssext  4228  nssss  4229
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
  Copyright terms: Public domain W3C validator