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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 GĂ©rard 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.
StepHypRef Expression
1 axgroth5 9223 . 2
2 simpl 457 . . . . . . . 8
32ralimi 2850 . . . . . . 7
4 pweq 4015 . . . . . . . . 9
54sseq1d 3530 . . . . . . . 8
65rspccv 3207 . . . . . . 7
73, 6syl 16 . . . . . 6
87anim2i 569 . . . . 5
983adant3 1016 . . . 4
10 pm3.35 587 . . . 4
11 vex 3112 . . . . 5
1211ssex 4596 . . . 4
139, 10, 123syl 20 . . 3
1413exlimiv 1722 . 2
151, 14ax-mp 5 1
 Colors of variables: wff setvar class 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
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