Theorem elpw2g 3910

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 3368	. 2 |- ( A e. ~P B -> A C_ B )
2		ssexg 3896	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
3		elpwg 3367	. . . . 5 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
4	3	biimparc 283	. . . 4 |- ( ( A C_ B /\ A e. _V ) -> A e. ~P B )
5	2, 4	syldan 266	. . 3 |- ( ( A C_ B /\ B e. V ) -> A e. ~P B )
6	5	expcom 109	. 2 |- ( B e. V -> ( A C_ B -> A e. ~P B ) )
7	1, 6	impbid2 131	1 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  elpw2  3911  pwnss  3912