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Theorem grothpw 9207
 Description: Derive the Axiom of Power Sets ax-pow 4615 from the Tarski-Grothendieck axiom ax-groth 9204. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4615 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpw
Distinct variable group:   ,,,

Proof of Theorem grothpw
StepHypRef Expression
1 axgroth5 9205 . . 3
2 simpl 457 . . . . . . . . 9
32ralimi 2836 . . . . . . . 8
4 pweq 4000 . . . . . . . . . 10
54sseq1d 3516 . . . . . . . . 9
65rspccv 3193 . . . . . . . 8
73, 6syl 16 . . . . . . 7
87anim2i 569 . . . . . 6
983adant3 1017 . . . . 5
10 pm3.35 587 . . . . 5
11 vex 3098 . . . . . 6
1211ssex 4581 . . . . 5
139, 10, 123syl 20 . . . 4
1413exlimiv 1709 . . 3
151, 14ax-mp 5 . 2
16 pwidg 4010 . . . . 5
17 pweq 4000 . . . . . . 7
1817eleq2d 2513 . . . . . 6
1918spcegv 3181 . . . . 5
2016, 19mpd 15 . . . 4
21 elex 3104 . . . . 5
2221exlimiv 1709 . . . 4
2320, 22impbii 188 . . 3
2411elpw2 4601 . . . . 5
25 pwss 4012 . . . . . 6
26 dfss2 3478 . . . . . . . 8
2726imbi1i 325 . . . . . . 7
2827albii 1627 . . . . . 6
2925, 28bitri 249 . . . . 5
3024, 29bitri 249 . . . 4
3130exbii 1654 . . 3
3223, 31bitri 249 . 2
3315, 32mpbi 208 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 368   wa 369   w3a 974  wal 1381   wceq 1383  wex 1599   wcel 1804  wral 2793  wrex 2794  cvv 3095   wss 3461  cpw 3997   class class class wbr 4437   cen 7515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-groth 9204 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-v 3097  df-in 3468  df-ss 3475  df-pw 3999 This theorem is referenced by: (None)
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