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Theorem grupw 9194
 Description: A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grupw

Proof of Theorem grupw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 9191 . . . . 5
21ibi 241 . . . 4
32simprd 463 . . 3
4 simp1 996 . . . 4
54ralimi 2850 . . 3
6 pweq 4015 . . . . 5
76eleq1d 2526 . . . 4
87rspccv 3207 . . 3
93, 5, 83syl 20 . 2
109imp 429 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  ~Pcpw 4012  {cpr 4031  U.cuni 4249  Trwtr 4545  rancrn 5005  (class class class)co 6296   cmap 7439   cgru 9189 This theorem is referenced by:  gruss  9195  grurn  9200  gruxp  9206  grumap  9207  gruwun  9212  intgru  9213  gruina  9217  grur1a  9218 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 This theorem depends on definitions:  df-bi 185  df-or 370  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-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-tr 4546  df-iota 5556  df-fv 5601  df-ov 6299  df-gru 9190
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