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Theorem wununi 8985
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 8983 . . . . 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 simp1 988 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  U. x  e.  U
)
76ralimi 2819 . . 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 4208 . . . 4  |-  ( x  =  A  ->  U. x  =  U. A )
109eleq1d 2523 . . 3  |-  ( x  =  A  ->  ( U. x  e.  U  <->  U. A  e.  U ) )
1110rspcv 3175 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   (/)c0 3746   ~Pcpw 3969   {cpr 3988   U.cuni 4200   Tr wtr 4494  WUnicwun 8979
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 1955  ax-ext 2432
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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-in 3444  df-ss 3451  df-uni 4201  df-tr 4495  df-wun 8981
This theorem is referenced by:  wunun  8989  wunint  8994  wundm  9007  wunrn  9008  wunfv  9011  intwun  9014  wuncval2  9026  wunstr  14312  wunfunc  14929  wunnat  14986  catcoppccl  15096  catcfuccl  15097  catcxpccl  15137
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