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Theorem wunpw 9115
Description: A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1  |-  ( ph  ->  U  e. WUni )
wununi.2  |-  ( ph  ->  A  e.  U )
Assertion
Ref Expression
wunpw  |-  ( ph  ->  ~P A  e.  U
)

Proof of Theorem wunpw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wununi.2 . 2  |-  ( ph  ->  A  e.  U )
2 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
3 iswun 9112 . . . . 5  |-  ( U  e. WUni  ->  ( U  e. WUni  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
43ibi 241 . . . 4  |-  ( U  e. WUni  ->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) )
54simp3d 1011 . . 3  |-  ( U  e. WUni  ->  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) )
6 simp2 998 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  ~P x  e.  U )
76ralimi 2797 . . 3  |-  ( A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  A. x  e.  U  ~P x  e.  U
)
82, 5, 73syl 18 . 2  |-  ( ph  ->  A. x  e.  U  ~P x  e.  U
)
9 pweq 3958 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
109eleq1d 2471 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  U  <->  ~P A  e.  U ) )
1110rspcv 3156 . 2  |-  ( A  e.  U  ->  ( A. x  e.  U  ~P x  e.  U  ->  ~P A  e.  U
) )
121, 8, 11sylc 59 1  |-  ( ph  ->  ~P A  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   (/)c0 3738   ~Pcpw 3955   {cpr 3974   U.cuni 4191   Tr wtr 4489  WUnicwun 9108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-in 3421  df-ss 3428  df-pw 3957  df-uni 4192  df-tr 4490  df-wun 9110
This theorem is referenced by:  wunss  9120  wunr1om  9127  wunxp  9132  wunpm  9133  intwun  9143  r1wunlim  9145  wuncval2  9155  wuncn  9577  wunfunc  15512  wunnat  15569  catcoppccl  15711  catcfuccl  15712  catcxpccl  15800
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