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Theorem pweqb 4230
Description: Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
Assertion
Ref Expression
pweqb  |-  ( A  =  B  <->  ~P A  =  ~P B )

Proof of Theorem pweqb
StepHypRef Expression
1 sspwb 4223 . . 3  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 sspwb 4223 . . 3  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2anbi12i 678 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A
) )
4 eqss 3194 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3194 . 2  |-  ( ~P A  =  ~P B  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A ) )
63, 4, 53bitr4i 268 1  |-  ( A  =  B  <->  ~P A  =  ~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152   ~Pcpw 3625
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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