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Theorem grupw 8670
 Description: A Grothendieck's 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 8667 . . . . 5
21ibi 233 . . . 4
32simprd 450 . . 3
4 simp1 957 . . . 4
54ralimi 2781 . . 3
6 pweq 3802 . . . . 5
76eleq1d 2502 . . . 4
87rspccv 3049 . . 3
93, 5, 83syl 19 . 2
109imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  cpw 3799  cpr 3815  cuni 4015   wtr 4302   crn 4879  (class class class)co 6081   cmap 7018  cgru 8665 This theorem is referenced by:  gruss  8671  grurn  8676  gruxp  8682  grumap  8683  gruwun  8688  intgru  8689  gruina  8693  grur1a  8694 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-tr 4303  df-iota 5418  df-fv 5462  df-ov 6084  df-gru 8666
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