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Theorem ssext 4204
Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssext  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ssext
StepHypRef Expression
1 ssextss 4203 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4203 . . 3  |-  ( B 
C_  A  <->  A. x
( x  C_  B  ->  x  C_  A )
)
31, 2anbi12i 457 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( A. x ( x  C_  A  ->  x  C_  B
)  /\  A. x
( x  C_  B  ->  x  C_  A )
) )
4 eqss 3162 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 albiim 1480 . 2  |-  ( A. x ( x  C_  A 
<->  x  C_  B )  <->  ( A. x ( x 
C_  A  ->  x  C_  B )  /\  A. x ( x  C_  B  ->  x  C_  A
) ) )
63, 4, 53bitr4i 211 1  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348    C_ wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587
This theorem is referenced by: (None)
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