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Theorem iswun 8514
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 4251 . . 3  |-  ( u  =  U  ->  ( Tr  u  <->  Tr  U )
)
2 neeq1 2560 . . 3  |-  ( u  =  U  ->  (
u  =/=  (/)  <->  U  =/=  (/) ) )
3 eleq2 2450 . . . . 5  |-  ( u  =  U  ->  ( U. x  e.  u  <->  U. x  e.  U ) )
4 eleq2 2450 . . . . 5  |-  ( u  =  U  ->  ( ~P x  e.  u  <->  ~P x  e.  U ) )
5 eleq2 2450 . . . . . 6  |-  ( u  =  U  ->  ( { x ,  y }  e.  u  <->  { x ,  y }  e.  U ) )
65raleqbi1dv 2857 . . . . 5  |-  ( u  =  U  ->  ( A. y  e.  u  { x ,  y }  e.  u  <->  A. y  e.  U  { x ,  y }  e.  U ) )
73, 4, 63anbi123d 1254 . . . 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 2857 . . 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 1254 . 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 8512 . 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 3029 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   (/)c0 3573   ~Pcpw 3744   {cpr 3760   U.cuni 3959   Tr wtr 4245  WUnicwun 8510
This theorem is referenced by:  wuntr  8515  wununi  8516  wunpw  8517  wunpr  8519  wun0  8528  intwun  8545  r1limwun  8546  wunex2  8548  tskwun  8594  gruwun  8623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-v 2903  df-in 3272  df-ss 3279  df-uni 3960  df-tr 4246  df-wun 8512
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