Theorem elpw2g 2727

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47.

Assertion

Ref	Expression
elpw2g	|- (B e. C -> (A e. P~B <-> A (_ B))

Proof of Theorem elpw2g

Step	Hyp	Ref	Expression
1		elpwi 2406	. 2 |- (A e. P~B -> A (_ B)
2		ssexg 2721	. . . 4 |- ((A (_ B /\ B e. C) -> A e. V)
3		elpwg 2405	. . . . 5 |- (A e. V -> (A e. P~B <-> A (_ B))
4	3	biimparc 419	. . . 4 |- ((A (_ B /\ A e. V) -> A e. P~B)
5	2, 4	syldan 467	. . 3 |- ((A (_ B /\ B e. C) -> A e. P~B)
6	5	expcom 374	. 2 |- (B e. C -> (A (_ B -> A e. P~B))
7	1, 6	impbid2 518	1 |- (B e. C -> (A e. P~B <-> A (_ B))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  elpw2 2728  cncfval 7264  uniopnt 7598  ntrval 7676  clsval 7677  neiss2 7716  neival 7717  lpval 7743  islp2 7747  blf 7844  iscau 7936  ump 10459  fillsb 10560  filint2 10574  filint2OLD 10575  efilcp2 10581  efilcp2OLD 10582  rcfpfil 10597  rcfpfilOLD 10598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402

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