Theorem elpwg 2409

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 2732.

Assertion

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

Proof of Theorem elpwg

Step	Hyp	Ref	Expression
1		eleq1 1537	. . 3 |- (x = A -> (x e. P~B <-> A e. P~B))
2		sseq1 2085	. . 3 |- (x = A -> (x (_ B <-> A (_ B))
3	1, 2	bibi12d 631	. 2 |- (x = A -> ((x e. P~B <-> x (_ B) <-> (A e. P~B <-> A (_ B)))
4		visset 1816	. . 3 |- x e. V
5	4	elpw 2408	. 2 |- (x e. P~B <-> x (_ B)
6	3, 5	vtoclg 1850	1 |- (A e. C -> (A e. P~B <-> A (_ B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  elpwi 2410  elpw2g 2732  pwel 2765  eldifpw 2916  elpwun 2917  elpwunsn 2918  r1rankid 4704  inpws1 10445  mapdiscn 10497  cnfilca 10562

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

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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