Theorem elpw2 2733

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 2732	. 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 960  Vcvv 1814   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  rankval2 4680  rankss 4698  aceq3lem 4742  bcthlem12 8007  ocvalt 9148  spanvalt 9294  hsupval2t 9295  sshjvalt 9315  sshjval3t 9321  dtopcl 10586

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406

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