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Theorem pweqb 4050
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 4043 . . 3  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
2 sspwb 4043 . . 3  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
31, 2anbi12i 448 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A
) )
4 eqss 3040 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3040 . 2  |-  ( ~P A  =  ~P B  <->  ( ~P A  C_  ~P B  /\  ~P B  C_  ~P A ) )
63, 4, 53bitr4i 210 1  |-  ( A  =  B  <->  ~P A  =  ~P B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    C_ wss 2999   ~Pcpw 3429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452
This theorem is referenced by: (None)
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