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Theorem ssext 4228
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 4227 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4227 . . 3  |-  ( B 
C_  A  <->  A. x
( x  C_  B  ->  x  C_  A )
)
31, 2anbi12i 678 . 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 3194 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 albiim 1598 . 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 268 1  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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