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Theorem eltskg 8625
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 3370 . . . . 5  |-  ( y  =  T  ->  ( ~P z  C_  y  <->  ~P z  C_  T ) )
2 rexeq 2905 . . . . 5  |-  ( y  =  T  ->  ( E. w  e.  y  ~P z  C_  w  <->  E. w  e.  T  ~P z  C_  w ) )
31, 2anbi12d 692 . . . 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 2912 . . 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 3802 . . . 4  |-  ( y  =  T  ->  ~P y  =  ~P T
)
6 breq2 4216 . . . . 5  |-  ( y  =  T  ->  (
z  ~~  y  <->  z  ~~  T ) )
7 eleq2 2497 . . . . 5  |-  ( y  =  T  ->  (
z  e.  y  <->  z  e.  T ) )
86, 7orbi12d 691 . . . 4  |-  ( y  =  T  ->  (
( z  ~~  y  \/  z  e.  y
)  <->  ( z  ~~  T  \/  z  e.  T ) ) )
95, 8raleqbidv 2916 . . 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 692 . 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 8624 . 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 3084 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212    ~~ cen 7106   Tarskictsk 8623
This theorem is referenced by:  eltsk2g  8626  tskpwss  8627  tsken  8629  grothtsk  8710
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-tsk 8624
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