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Theorem eltskg 9124
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 3526 . . . . 5  |-  ( y  =  T  ->  ( ~P z  C_  y  <->  ~P z  C_  T ) )
2 rexeq 3059 . . . . 5  |-  ( y  =  T  ->  ( E. w  e.  y  ~P z  C_  w  <->  E. w  e.  T  ~P z  C_  w ) )
31, 2anbi12d 710 . . . 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 3066 . . 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 4013 . . . 4  |-  ( y  =  T  ->  ~P y  =  ~P T
)
6 breq2 4451 . . . . 5  |-  ( y  =  T  ->  (
z  ~~  y  <->  z  ~~  T ) )
7 eleq2 2540 . . . . 5  |-  ( y  =  T  ->  (
z  e.  y  <->  z  e.  T ) )
86, 7orbi12d 709 . . . 4  |-  ( y  =  T  ->  (
( z  ~~  y  \/  z  e.  y
)  <->  ( z  ~~  T  \/  z  e.  T ) ) )
95, 8raleqbidv 3072 . . 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 710 . 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 9123 . 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 3252 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 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447    ~~ cen 7510   Tarskictsk 9122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-tsk 9123
This theorem is referenced by:  eltsk2g  9125  tskpwss  9126  tsken  9128  grothtsk  9209
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