Theorem pweqb 4308

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 4301	. . 3 |- ( A C_ B <-> ~P A C_ ~P B )
2		sspwb 4301	. . 3 |- ( B C_ A <-> ~P B C_ ~P A )
3	1, 2	anbi12i 460	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
4		eqss 3239	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3239	. 2 |- ( ~P A = ~P B <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
6	3, 4, 5	3bitr4i 212	1 |- ( A = B <-> ~P A = ~P B )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   C_ wss 3197   ~Pcpw 3649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by: (None)