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Theorem wunpw 9081
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 9078 . . . . 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 1010 . . 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 997 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  ~P x  e.  U )
76ralimi 2857 . . 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 20 . 2  |-  ( ph  ->  A. x  e.  U  ~P x  e.  U
)
9 pweq 4013 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
109eleq1d 2536 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  U  <->  ~P A  e.  U ) )
1110rspcv 3210 . 2  |-  ( A  e.  U  ->  ( A. x  e.  U  ~P x  e.  U  ->  ~P A  e.  U
) )
121, 8, 11sylc 60 1  |-  ( ph  ->  ~P A  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   (/)c0 3785   ~Pcpw 4010   {cpr 4029   U.cuni 4245   Tr wtr 4540  WUnicwun 9074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-tr 4541  df-wun 9076
This theorem is referenced by:  wunss  9086  wunr1om  9093  wunxp  9098  wunpm  9099  intwun  9109  r1wunlim  9111  wuncval2  9121  wuncn  9543  wunfunc  15119  wunnat  15176  catcoppccl  15286  catcfuccl  15287  catcxpccl  15327
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