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Theorem eltskg 8372
Description: Properties of a Tarski's class. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
eltskg  |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  /\  A. z  e.  ~P  T
( z  ~~  T  \/  z  e.  T
) ) ) )
Distinct variable group:    w, T, z
Allowed substitution hints:    V( z, w)

Proof of Theorem eltskg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sseq2 3200 . . . . 5  |-  ( y  =  T  ->  ( ~P z  C_  y  <->  ~P z  C_  T ) )
2 rexeq 2737 . . . . 5  |-  ( y  =  T  ->  ( E. w  e.  y  ~P z  C_  w  <->  E. w  e.  T  ~P z  C_  w ) )
31, 2anbi12d 691 . . . 4  |-  ( y  =  T  ->  (
( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  <->  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) ) )
43raleqbi1dv 2744 . . 3  |-  ( y  =  T  ->  ( A. z  e.  y 
( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  <->  A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) ) )
5 pweq 3628 . . . 4  |-  ( y  =  T  ->  ~P y  =  ~P T
)
6 breq2 4027 . . . . 5  |-  ( y  =  T  ->  (
z  ~~  y  <->  z  ~~  T ) )
7 eleq2 2344 . . . . 5  |-  ( y  =  T  ->  (
z  e.  y  <->  z  e.  T ) )
86, 7orbi12d 690 . . . 4  |-  ( y  =  T  ->  (
( z  ~~  y  \/  z  e.  y
)  <->  ( z  ~~  T  \/  z  e.  T ) ) )
95, 8raleqbidv 2748 . . 3  |-  ( y  =  T  ->  ( A. z  e.  ~P  y ( z  ~~  y  \/  z  e.  y )  <->  A. z  e.  ~P  T ( z 
~~  T  \/  z  e.  T ) ) )
104, 9anbi12d 691 . 2  |-  ( y  =  T  ->  (
( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  /\  A. z  e.  ~P  T
( z  ~~  T  \/  z  e.  T
) ) ) )
11 df-tsk 8371 . 2  |-  Tarski  =  {
y  |  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y ( z  ~~  y  \/  z  e.  y ) ) }
1210, 11elab2g 2916 1  |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  /\  A. z  e.  ~P  T
( z  ~~  T  \/  z  e.  T
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    ~~ cen 6860   Tarskictsk 8370
This theorem is referenced by:  eltsk2g  8373  tskpwss  8374  tsken  8376  grothtsk  8457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-tsk 8371
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