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Theorem wununi 9073
Description: A weak universe is closed under union. (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
wununi  |-  ( ph  ->  U. A  e.  U
)

Proof of Theorem wununi
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 9071 . . . . 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 1008 . . 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 simp1 994 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  U. x  e.  U
)
76ralimi 2847 . . 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  U. x  e.  U
)
82, 5, 73syl 20 . 2  |-  ( ph  ->  A. x  e.  U  U. x  e.  U
)
9 unieq 4243 . . . 4  |-  ( x  =  A  ->  U. x  =  U. A )
109eleq1d 2523 . . 3  |-  ( x  =  A  ->  ( U. x  e.  U  <->  U. A  e.  U ) )
1110rspcv 3203 . 2  |-  ( A  e.  U  ->  ( A. x  e.  U  U. x  e.  U  ->  U. A  e.  U
) )
121, 8, 11sylc 60 1  |-  ( ph  ->  U. A  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   (/)c0 3783   ~Pcpw 3999   {cpr 4018   U.cuni 4235   Tr wtr 4532  WUnicwun 9067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-uni 4236  df-tr 4533  df-wun 9069
This theorem is referenced by:  wunun  9077  wunint  9082  wundm  9095  wunrn  9096  wunfv  9099  intwun  9102  wuncval2  9114  wunstr  14735  wunfunc  15387  wunnat  15444  catcoppccl  15586  catcfuccl  15587  catcxpccl  15675
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