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Theorem ssextss 4249
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 4245 . 2  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 dfss2 3168 . 2  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
3 vex 2763 . . . . 5  |-  x  e. 
_V
43elpw 3607 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3607 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5imbi12i 239 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  ~P B
)  <->  ( x  C_  A  ->  x  C_  B
) )
76albii 1481 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  <->  A. x
( x  C_  A  ->  x  C_  B )
)
81, 2, 73bitri 206 1  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362    e. wcel 2164    C_ wss 3153   ~Pcpw 3601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624
This theorem is referenced by:  ssext  4250  nssssr  4251
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