Theorem elpw2 2728

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

Hypothesis

Ref	Expression
elpw2.1	|- B e. V

Assertion

Ref	Expression
elpw2	|- (A e. P~B <-> A (_ B)

Proof of Theorem elpw2

Step	Hyp	Ref	Expression
1		elpw2.1	. 2 |- B e. V
2		elpw2g 2727	. 2 |- (B e. V -> (A e. P~B <-> A (_ B))
3	1, 2	ax-mp 7	1 |- (A e. P~B <-> A (_ B)

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

This theorem is referenced by:  rankval2 4670  rankss 4688  aceq3lem 4732  bcthlem12 8010  ocvalt 9153  spanvalt 9299  hsupval2t 9300  sshjvalt 9320  sshjval3t 9326

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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