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Theorem pweqb 4168
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 4161 . . 3  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 sspwb 4161 . . 3  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2anbi12i 681 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A
) )
4 eqss 3136 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3136 . 2  |-  ( ~P A  =  ~P B  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A ) )
63, 4, 53bitr4i 270 1  |-  ( A  =  B  <->  ~P A  =  ~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    C_ wss 3094   ~Pcpw 3566
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-pw 3568  df-sn 3587  df-pr 3588
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