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Theorem grupw 8417
Description: A Grothendieck's universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grupw  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )

Proof of Theorem grupw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 8414 . . . . 5  |-  ( U  e.  Univ  ->  ( U  e.  Univ  <->  ( Tr  U  /\  A. y  e.  U  ( ~P y  e.  U  /\  A. x  e.  U  { y ,  x }  e.  U  /\  A. x  e.  ( U  ^m  y ) U. ran  x  e.  U ) ) ) )
21ibi 232 . . . 4  |-  ( U  e.  Univ  ->  ( Tr  U  /\  A. y  e.  U  ( ~P y  e.  U  /\  A. x  e.  U  {
y ,  x }  e.  U  /\  A. x  e.  ( U  ^m  y
) U. ran  x  e.  U ) ) )
32simprd 449 . . 3  |-  ( U  e.  Univ  ->  A. y  e.  U  ( ~P y  e.  U  /\  A. x  e.  U  {
y ,  x }  e.  U  /\  A. x  e.  ( U  ^m  y
) U. ran  x  e.  U ) )
4 simp1 955 . . . 4  |-  ( ( ~P y  e.  U  /\  A. x  e.  U  { y ,  x }  e.  U  /\  A. x  e.  ( U  ^m  y ) U. ran  x  e.  U )  ->  ~P y  e.  U )
54ralimi 2618 . . 3  |-  ( A. y  e.  U  ( ~P y  e.  U  /\  A. x  e.  U  { y ,  x }  e.  U  /\  A. x  e.  ( U  ^m  y ) U. ran  x  e.  U )  ->  A. y  e.  U  ~P y  e.  U
)
6 pweq 3628 . . . . 5  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76eleq1d 2349 . . . 4  |-  ( y  =  A  ->  ( ~P y  e.  U  <->  ~P A  e.  U ) )
87rspccv 2881 . . 3  |-  ( A. y  e.  U  ~P y  e.  U  ->  ( A  e.  U  ->  ~P A  e.  U
) )
93, 5, 83syl 18 . 2  |-  ( U  e.  Univ  ->  ( A  e.  U  ->  ~P A  e.  U )
)
109imp 418 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ~Pcpw 3625   {cpr 3641   U.cuni 3827   Tr wtr 4113   ran crn 4690  (class class class)co 5858    ^m cmap 6772   Univcgru 8412
This theorem is referenced by:  gruss  8418  grurn  8423  gruxp  8429  grumap  8430  gruwun  8435  intgru  8436  gruina  8440  grur1a  8441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-iota 5219  df-fv 5263  df-ov 5861  df-gru 8413
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