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Theorem iswun 9080
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 4532 . . 3  |-  ( u  =  U  ->  ( Tr  u  <->  Tr  U )
)
2 neeq1 2722 . . 3  |-  ( u  =  U  ->  (
u  =/=  (/)  <->  U  =/=  (/) ) )
3 eleq2 2514 . . . . 5  |-  ( u  =  U  ->  ( U. x  e.  u  <->  U. x  e.  U ) )
4 eleq2 2514 . . . . 5  |-  ( u  =  U  ->  ( ~P x  e.  u  <->  ~P x  e.  U ) )
5 eleq2 2514 . . . . . 6  |-  ( u  =  U  ->  ( { x ,  y }  e.  u  <->  { x ,  y }  e.  U ) )
65raleqbi1dv 3046 . . . . 5  |-  ( u  =  U  ->  ( A. y  e.  u  { x ,  y }  e.  u  <->  A. y  e.  U  { x ,  y }  e.  U ) )
73, 4, 63anbi123d 1298 . . . 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 3046 . . 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 1298 . 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 9078 . 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 3232 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   (/)c0 3767   ~Pcpw 3993   {cpr 4012   U.cuni 4230   Tr wtr 4526  WUnicwun 9076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-v 3095  df-in 3465  df-ss 3472  df-uni 4231  df-tr 4527  df-wun 9078
This theorem is referenced by:  wuntr  9081  wununi  9082  wunpw  9083  wunpr  9085  wun0  9094  intwun  9111  r1limwun  9112  wunex2  9114  tskwun  9160  gruwun  9189  pwinfi2  37412
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