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Theorem pweqb 4153
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 4146 . . 3  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 sspwb 4146 . . 3  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2anbi12i 456 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A
) )
4 eqss 3117 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3117 . 2  |-  ( ~P A  =  ~P B  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A ) )
63, 4, 53bitr4i 211 1  |-  ( A  =  B  <->  ~P A  =  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1332    C_ wss 3076   ~Pcpw 3515
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538
This theorem is referenced by: (None)
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