Theorem pweqb 3978

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

Step	Hyp	Ref	Expression
1		sspwb 3971	. . 3 |- ( A C_ B <-> ~P A C_ ~P B )
2		sspwb 3971	. . 3 |- ( B C_ A <-> ~P B C_ ~P A )
3	1, 2	anbi12i 447	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
4		eqss 3014	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3014	. 2 |- ( ~P A = ~P B <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
6	3, 4, 5	3bitr4i 210	1 |- ( A = B <-> ~P A = ~P B )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   C_ wss 2973   ~Pcpw 3382

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by: (None)