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Theorem iswun 8571
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 4300 . . 3  |-  ( u  =  U  ->  ( Tr  u  <->  Tr  U )
)
2 neeq1 2606 . . 3  |-  ( u  =  U  ->  (
u  =/=  (/)  <->  U  =/=  (/) ) )
3 eleq2 2496 . . . . 5  |-  ( u  =  U  ->  ( U. x  e.  u  <->  U. x  e.  U ) )
4 eleq2 2496 . . . . 5  |-  ( u  =  U  ->  ( ~P x  e.  u  <->  ~P x  e.  U ) )
5 eleq2 2496 . . . . . 6  |-  ( u  =  U  ->  ( { x ,  y }  e.  u  <->  { x ,  y }  e.  U ) )
65raleqbi1dv 2904 . . . . 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 2904 . . 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 8569 . 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 3076 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   (/)c0 3620   ~Pcpw 3791   {cpr 3807   U.cuni 4007   Tr wtr 4294  WUnicwun 8567
This theorem is referenced by:  wuntr  8572  wununi  8573  wunpw  8574  wunpr  8576  wun0  8585  intwun  8602  r1limwun  8603  wunex2  8605  tskwun  8651  gruwun  8680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-wun 8569
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