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Theorem grupw 8433
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 8430 . . . . 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 2631 . . 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 3641 . . . . 5  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76eleq1d 2362 . . . 4  |-  ( y  =  A  ->  ( ~P y  e.  U  <->  ~P A  e.  U ) )
87rspccv 2894 . . 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 1632    e. wcel 1696   A.wral 2556   ~Pcpw 3638   {cpr 3654   U.cuni 3843   Tr wtr 4129   ran crn 4706  (class class class)co 5874    ^m cmap 6788   Univcgru 8428
This theorem is referenced by:  gruss  8434  grurn  8439  gruxp  8445  grumap  8446  gruwun  8451  intgru  8452  gruina  8456  grur1a  8457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-tr 4130  df-iota 5235  df-fv 5279  df-ov 5877  df-gru 8429
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