Theorem elpw2g 3931

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Assertion

Ref	Expression
elpw2g	|- ( B e. V -> ( A e. ~P B <-> A C_ B ) )

Proof of Theorem elpw2g

Step	Hyp	Ref	Expression
1		elpwi 3391	. 2 |- ( A e. ~P B -> A C_ B )
2		ssexg 3917	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
3		elpwg 3390	. . . . 5 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
4	3	biimparc 293	. . . 4 |- ( ( A C_ B /\ A e. _V ) -> A e. ~P B )
5	2, 4	syldan 276	. . 3 |- ( ( A C_ B /\ B e. V ) -> A e. ~P B )
6	5	expcom 114	. 2 |- ( B e. V -> ( A C_ B -> A e. ~P B ) )
7	1, 6	impbid2 141	1 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   _Vcvv 2601   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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

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

This theorem is referenced by:  elpw2  3932  pwnss  3933