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Theorem ssextss 3983
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 3979 . 2  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 dfss2 2989 . 2  |-  ( ~P A  C_  ~P B  <->  A. x ( x  e. 
~P A  ->  x  e.  ~P B ) )
3 vex 2605 . . . . 5  |-  x  e. 
_V
43elpw 3396 . . . 4  |-  ( x  e.  ~P A  <->  x  C_  A
)
53elpw 3396 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
64, 5imbi12i 237 . . 3  |-  ( ( x  e.  ~P A  ->  x  e.  ~P B
)  <->  ( x  C_  A  ->  x  C_  B
) )
76albii 1400 . 2  |-  ( A. x ( x  e. 
~P A  ->  x  e.  ~P B )  <->  A. x
( x  C_  A  ->  x  C_  B )
)
81, 2, 73bitri 204 1  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    e. wcel 1434    C_ wss 2974   ~Pcpw 3390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412
This theorem is referenced by:  ssext  3984  nssssr  3985
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