Theorem pweqb 4208

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 4201	. . 3 |- ( A C_ B <-> ~P A C_ ~P B )
2		sspwb 4201	. . 3 |- ( B C_ A <-> ~P B C_ ~P A )
3	1, 2	anbi12i 457	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
4		eqss 3162	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3162	. 2 |- ( ~P A = ~P B <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
6	3, 4, 5	3bitr4i 211	1 |- ( A = B <-> ~P A = ~P B )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)