Theorem elpwg 2405

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 2727.

Assertion

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

Proof of Theorem elpwg

Step	Hyp	Ref	Expression
1		eleq1 1534	. . 3 |- (x = A -> (x e. P~B <-> A e. P~B))
2		sseq1 2082	. . 3 |- (x = A -> (x (_ B <-> A (_ B))
3	1, 2	bibi12d 629	. 2 |- (x = A -> ((x e. P~B <-> x (_ B) <-> (A e. P~B <-> A (_ B)))
4		visset 1813	. . 3 |- x e. V
5	4	elpw 2404	. 2 |- (x e. P~B <-> x (_ B)
6	3, 5	vtoclg 1847	1 |- (A e. C -> (A e. P~B <-> A (_ B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  elpwi 2406  elpw2g 2727  pwel 2759  eldifpw 2910  elpwun 2911  elpwunsn 2912  r1rankid 4694  inpws1 10455  mapdiscn 10511  cnfilca 10583  cnfilcaOLD 10584

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

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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