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Theorem ssext 3985
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 3984 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 3984 . . 3  |-  ( B 
C_  A  <->  A. x
( x  C_  B  ->  x  C_  A )
)
31, 2anbi12i 441 . 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 2988 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 albiim 1392 . 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 205 1  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by: (None)
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