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Theorem wunpw 8972
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 8969 . . . . 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 1002 . . 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 989 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  ~P x  e.  U )
76ralimi 2809 . . 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 3958 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
109eleq1d 2519 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  U  <->  ~P A  e.  U ) )
1110rspcv 3162 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   (/)c0 3732   ~Pcpw 3955   {cpr 3974   U.cuni 4186   Tr wtr 4480  WUnicwun 8965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-v 3067  df-in 3430  df-ss 3437  df-pw 3957  df-uni 4187  df-tr 4481  df-wun 8967
This theorem is referenced by:  wunss  8977  wunr1om  8984  wunxp  8989  wunpm  8990  intwun  9000  r1wunlim  9002  wuncval2  9012  wuncn  9435  wunfunc  14908  wunnat  14965  catcoppccl  15075  catcfuccl  15076  catcxpccl  15116
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