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Theorem ssextss 4264
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 4260 . 2  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 dfss2 3203 . 2  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
3 vex 2825 . . . . 5  |-  x  e. 
_V
43elpw 3665 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3665 . . . 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 1557 . 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 1531    e. wcel 1701    C_ wss 3186   ~Pcpw 3659
This theorem is referenced by:  ssext  4265  nssss  4266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-pw 3661  df-sn 3680  df-pr 3681
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