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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  |-  ( ( 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 8667 . . . . 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 233 . . . 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 450 . . 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 957 . . . 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 2781 . . 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 3802 . . . . 5  |-  ( y  =  A  ->  ~P y  =  ~P A
)
76eleq1d 2502 . . . 4  |-  ( y  =  A  ->  ( ~P y  e.  U  <->  ~P A  e.  U ) )
87rspccv 3049 . . 3  |-  ( A. y  e.  U  ~P y  e.  U  ->  ( A  e.  U  ->  ~P A  e.  U
) )
93, 5, 83syl 19 . 2  |-  ( U  e.  Univ  ->  ( A  e.  U  ->  ~P A  e.  U )
)
109imp 419 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ~Pcpw 3799   {cpr 3815   U.cuni 4015   Tr wtr 4302   ran crn 4879  (class class class)co 6081    ^m cmap 7018   Univcgru 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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