Theorem grothpwex 9226

Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9222. Note that ax-pow 4630 is not used by the proof. Use axpweq 4629 to obtain ax-pow 4630. Use pwex 4635 or pwexg 4636 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
grothpwex

Proof of Theorem grothpwex

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axgroth5 9223	. 2
2		simpl 457	. . . . . . . 8
3	2	ralimi 2850	. . . . . . 7
4		pweq 4015	. . . . . . . . 9
5	4	sseq1d 3530	. . . . . . . 8
6	5	rspccv 3207	. . . . . . 7
7	3, 6	syl 16	. . . . . 6
8	7	anim2i 569	. . . . 5
9	8	3adant3 1016	. . . 4
10		pm3.35 587	. . . 4
11		vex 3112	. . . . 5
12	11	ssex 4596	. . . 4
13	9, 10, 12	3syl 20	. . 3
14	13	exlimiv 1722	. 2
15	1, 14	ax-mp 5	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  /\w3a 973  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  ~Pcpw 4012   class class class wbr 4452  cen 7533

This theorem is referenced by:  isrnsigaOLD  28112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-groth 9222

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014