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Theorem grupw 9121
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 9118 . . . . 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 461 . . 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 995 . . . 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 2794 . . 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 3955 . . . . 5  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76eleq1d 2469 . . . 4  |-  ( y  =  A  ->  ( ~P y  e.  U  <->  ~P A  e.  U ) )
87rspccv 3154 . . 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 427 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   ~Pcpw 3952   {cpr 3971   U.cuni 4188   Tr wtr 4486   ran crn 4941  (class class class)co 6232    ^m cmap 7375   Univcgru 9116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-tr 4487  df-iota 5487  df-fv 5531  df-ov 6235  df-gru 9117
This theorem is referenced by:  gruss  9122  grurn  9127  gruxp  9133  grumap  9134  gruwun  9139  intgru  9140  gruina  9144  grur1a  9145
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