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Theorem grupw 9164
Description: A Grothendieck 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 9161 . . . . 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 241 . . . 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 463 . . 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 991 . . . 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 2852 . . 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 4008 . . . . 5  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76eleq1d 2531 . . . 4  |-  ( y  =  A  ->  ( ~P y  e.  U  <->  ~P A  e.  U ) )
87rspccv 3206 . . 3  |-  ( A. y  e.  U  ~P y  e.  U  ->  ( A  e.  U  ->  ~P A  e.  U
) )
93, 5, 83syl 20 . 2  |-  ( U  e.  Univ  ->  ( A  e.  U  ->  ~P A  e.  U )
)
109imp 429 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   ~Pcpw 4005   {cpr 4024   U.cuni 4240   Tr wtr 4535   ran crn 4995  (class class class)co 6277    ^m cmap 7412   Univcgru 9159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-tr 4536  df-iota 5544  df-fv 5589  df-ov 6280  df-gru 9160
This theorem is referenced by:  gruss  9165  grurn  9170  gruxp  9176  grumap  9177  gruwun  9182  intgru  9183  gruina  9187  grur1a  9188
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