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Theorem ssextss 4417
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 4413 . 2  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 dfss2 3337 . 2  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
3 vex 2959 . . . . 5  |-  x  e. 
_V
43elpw 3805 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3805 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5imbi12i 317 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  ~P B
)  <->  ( x  C_  A  ->  x  C_  B
) )
76albii 1575 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  <->  A. x
( x  C_  A  ->  x  C_  B )
)
81, 2, 73bitri 263 1  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725    C_ wss 3320   ~Pcpw 3799
This theorem is referenced by:  ssext  4418  nssss  4419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821
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