Theorem pwexg 1807

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17.

Assertion

Ref	Expression
pwexg	|- (A e. B -> P~A e. V)

Proof of Theorem pwexg

Step	Hyp	Ref	Expression
1		pweq 1800	. . 3 |- (x = A -> P~x = P~A)
2	1	eleq1d 1155	. 2 |- (x = A -> (P~x e. V <-> P~A e. V))
3		visset 1350	. . 3 |- x e. V
4	3	pwex 1806	. 2 |- P~x e. V
5	2, 4	vtoclg 1383	1 |- (A e. B -> P~A e. V)

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  Vcvv 1348  P~cpw 1798

This theorem is referenced by:  uniexb 1962  mapex 3261  canth3 3656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799

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