Theorem pweqb 4140

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 4133	. . 3 |- ( A C_ B <-> ~P A C_ ~P B )
2		sspwb 4133	. . 3 |- ( B C_ A <-> ~P B C_ ~P A )
3	1, 2	anbi12i 455	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( ~P A C_ ~P B /\ ~P B C_ ~P A ) )
4		eqss 3107	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3107	. 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 1331   C_ wss 3066   ~Pcpw 3505

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528

This theorem is referenced by: (None)