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Theorem eltskg 9078
Description: Properties of a Tarski 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 3463 . . . . 5  |-  ( y  =  T  ->  ( ~P z  C_  y  <->  ~P z  C_  T ) )
2 rexeq 3004 . . . . 5  |-  ( y  =  T  ->  ( E. w  e.  y  ~P z  C_  w  <->  E. w  e.  T  ~P z  C_  w ) )
31, 2anbi12d 709 . . . 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 3011 . . 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 3957 . . . 4  |-  ( y  =  T  ->  ~P y  =  ~P T
)
6 breq2 4398 . . . . 5  |-  ( y  =  T  ->  (
z  ~~  y  <->  z  ~~  T ) )
7 eleq2 2475 . . . . 5  |-  ( y  =  T  ->  (
z  e.  y  <->  z  e.  T ) )
86, 7orbi12d 708 . . . 4  |-  ( y  =  T  ->  (
( z  ~~  y  \/  z  e.  y
)  <->  ( z  ~~  T  \/  z  e.  T ) ) )
95, 8raleqbidv 3017 . . 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 709 . 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 9077 . 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 3197 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    C_ wss 3413   ~Pcpw 3954   class class class wbr 4394    ~~ cen 7471   Tarskictsk 9076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-tsk 9077
This theorem is referenced by:  eltsk2g  9079  tskpwss  9080  tsken  9082  grothtsk  9163
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