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Theorem iswun 9071
Description: Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
iswun  |-  ( U  e.  V  ->  ( 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 ) ) ) )
Distinct variable group:    x, y, U
Allowed substitution hints:    V( x, y)

Proof of Theorem iswun
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 treq 4539 . . 3  |-  ( u  =  U  ->  ( Tr  u  <->  Tr  U )
)
2 neeq1 2741 . . 3  |-  ( u  =  U  ->  (
u  =/=  (/)  <->  U  =/=  (/) ) )
3 eleq2 2533 . . . . 5  |-  ( u  =  U  ->  ( U. x  e.  u  <->  U. x  e.  U ) )
4 eleq2 2533 . . . . 5  |-  ( u  =  U  ->  ( ~P x  e.  u  <->  ~P x  e.  U ) )
5 eleq2 2533 . . . . . 6  |-  ( u  =  U  ->  ( { x ,  y }  e.  u  <->  { x ,  y }  e.  U ) )
65raleqbi1dv 3059 . . . . 5  |-  ( u  =  U  ->  ( A. y  e.  u  { x ,  y }  e.  u  <->  A. y  e.  U  { x ,  y }  e.  U ) )
73, 4, 63anbi123d 1294 . . . 4  |-  ( u  =  U  ->  (
( U. x  e.  u  /\  ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u )  <->  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) )
87raleqbi1dv 3059 . . 3  |-  ( u  =  U  ->  ( 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  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) )
91, 2, 83anbi123d 1294 . 2  |-  ( u  =  U  ->  (
( Tr  u  /\  u  =/=  (/)  /\  A. x  e.  u  ( U. x  e.  u  /\  ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u ) )  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
10 df-wun 9069 . 2  |- WUni  =  {
u  |  ( Tr  u  /\  u  =/=  (/)  /\  A. x  e.  u  ( U. x  e.  u  /\  ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u ) ) }
119, 10elab2g 3245 1  |-  ( U  e.  V  ->  ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   (/)c0 3778   ~Pcpw 4003   {cpr 4022   U.cuni 4238   Tr wtr 4533  WUnicwun 9067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-v 3108  df-in 3476  df-ss 3483  df-uni 4239  df-tr 4534  df-wun 9069
This theorem is referenced by:  wuntr  9072  wununi  9073  wunpw  9074  wunpr  9076  wun0  9085  intwun  9102  r1limwun  9103  wunex2  9105  tskwun  9151  gruwun  9180
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