MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iswun Structured version   Unicode version

Theorem iswun 8958
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 4475 . . 3  |-  ( u  =  U  ->  ( Tr  u  <->  Tr  U )
)
2 neeq1 2726 . . 3  |-  ( u  =  U  ->  (
u  =/=  (/)  <->  U  =/=  (/) ) )
3 eleq2 2521 . . . . 5  |-  ( u  =  U  ->  ( U. x  e.  u  <->  U. x  e.  U ) )
4 eleq2 2521 . . . . 5  |-  ( u  =  U  ->  ( ~P x  e.  u  <->  ~P x  e.  U ) )
5 eleq2 2521 . . . . . 6  |-  ( u  =  U  ->  ( { x ,  y }  e.  u  <->  { x ,  y }  e.  U ) )
65raleqbi1dv 3007 . . . . 5  |-  ( u  =  U  ->  ( A. y  e.  u  { x ,  y }  e.  u  <->  A. y  e.  U  { x ,  y }  e.  U ) )
73, 4, 63anbi123d 1290 . . . 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 3007 . . 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 1290 . 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 8956 . 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 3191 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 965    = wceq 1370    e. wcel 1757    =/= wne 2641   A.wral 2792   (/)c0 3721   ~Pcpw 3944   {cpr 3963   U.cuni 4175   Tr wtr 4469  WUnicwun 8954
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 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
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 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-v 3056  df-in 3419  df-ss 3426  df-uni 4176  df-tr 4470  df-wun 8956
This theorem is referenced by:  wuntr  8959  wununi  8960  wunpw  8961  wunpr  8963  wun0  8972  intwun  8989  r1limwun  8990  wunex2  8992  tskwun  9038  gruwun  9067
  Copyright terms: Public domain W3C validator