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Theorem grothpw 8693
 Description: Derive the Axiom of Power Sets ax-pow 4369 from the Tarski-Grothendieck axiom ax-groth 8690. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4369 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.)
Assertion
Ref Expression
grothpw
Distinct variable group:   ,,,

Proof of Theorem grothpw
StepHypRef Expression
1 axgroth5 8691 . . 3
2 simpl 444 . . . . . . . . 9
32ralimi 2773 . . . . . . . 8
4 pweq 3794 . . . . . . . . . 10
54sseq1d 3367 . . . . . . . . 9
65rspccv 3041 . . . . . . . 8
73, 6syl 16 . . . . . . 7
87anim2i 553 . . . . . 6
983adant3 977 . . . . 5
10 pm3.35 571 . . . . 5
11 vex 2951 . . . . . 6
1211ssex 4339 . . . . 5
139, 10, 123syl 19 . . . 4
1413exlimiv 1644 . . 3
151, 14ax-mp 8 . 2
16 pwidg 3803 . . . . 5
17 pweq 3794 . . . . . . 7
1817eleq2d 2502 . . . . . 6
1918spcegv 3029 . . . . 5
2016, 19mpd 15 . . . 4
21 elex 2956 . . . . 5
2221exlimiv 1644 . . . 4
2320, 22impbii 181 . . 3
2411elpw2 4356 . . . . 5
25 pwss 3805 . . . . . 6
26 dfss2 3329 . . . . . . . 8
2726imbi1i 316 . . . . . . 7
2827albii 1575 . . . . . 6
2925, 28bitri 241 . . . . 5
3024, 29bitri 241 . . . 4
3130exbii 1592 . . 3
3223, 31bitri 241 . 2
3315, 32mpbi 200 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   w3a 936  wal 1549  wex 1550   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   wss 3312  cpw 3791   class class class wbr 4204   cen 7098 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-groth 8690 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793
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