MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grothpw Unicode version

Theorem grothpw 9225
Description: Derive the Axiom of Power Sets ax-pow 4630 from the Tarski-Grothendieck axiom ax-groth 9222. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4630 is not used by the proof. (Contributed by GĂ©rard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpw
Distinct variable group:   , , ,

Proof of Theorem grothpw
StepHypRef Expression
1 axgroth5 9223 . . 3
2 simpl 457 . . . . . . . . 9
32ralimi 2850 . . . . . . . 8
4 pweq 4015 . . . . . . . . . 10
54sseq1d 3530 . . . . . . . . 9
65rspccv 3207 . . . . . . . 8
73, 6syl 16 . . . . . . 7
87anim2i 569 . . . . . 6
983adant3 1016 . . . . 5
10 pm3.35 587 . . . . 5
11 vex 3112 . . . . . 6
1211ssex 4596 . . . . 5
139, 10, 123syl 20 . . . 4
1413exlimiv 1722 . . 3
151, 14ax-mp 5 . 2
16 pwidg 4025 . . . . 5
17 pweq 4015 . . . . . . 7
1817eleq2d 2527 . . . . . 6
1918spcegv 3195 . . . . 5
2016, 19mpd 15 . . . 4
21 elex 3118 . . . . 5
2221exlimiv 1722 . . . 4
2320, 22impbii 188 . . 3
2411elpw2 4616 . . . . 5
25 pwss 4027 . . . . . 6
26 dfss2 3492 . . . . . . . 8
2726imbi1i 325 . . . . . . 7
2827albii 1640 . . . . . 6
2925, 28bitri 249 . . . . 5
3024, 29bitri 249 . . . 4
3130exbii 1667 . . 3
3223, 31bitri 249 . 2
3315, 32mpbi 208 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  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 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
  Copyright terms: Public domain W3C validator