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Theorem eltsk2g 8628
Description: Properties of a Tarski's class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
eltsk2g  |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
)  /\  A. z  e.  ~P  T ( z 
~~  T  \/  z  e.  T ) ) ) )
Distinct variable group:    z, T
Allowed substitution hint:    V( z)

Proof of Theorem eltsk2g
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eltskg 8627 . 2  |-  ( 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
) ) ) )
2 nfra1 2758 . . . . . . 7  |-  F/ z A. z  e.  T  ~P z  C_  T
3 pweq 3804 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ~P z  =  ~P w
)
43sseq1d 3377 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( ~P z  C_  T  <->  ~P w  C_  T ) )
54rspccva 3053 . . . . . . . . . 10  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  w  e.  T )  ->  ~P w  C_  T
)
65adantlr 697 . . . . . . . . 9  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ~P w  C_  T )
7 vex 2961 . . . . . . . . . . . 12  |-  z  e. 
_V
87pwex 4384 . . . . . . . . . . 11  |-  ~P z  e.  _V
98elpw 3807 . . . . . . . . . 10  |-  ( ~P z  e.  ~P w  <->  ~P z  C_  w )
10 ssel 3344 . . . . . . . . . 10  |-  ( ~P w  C_  T  ->  ( ~P z  e.  ~P w  ->  ~P z  e.  T ) )
119, 10syl5bir 211 . . . . . . . . 9  |-  ( ~P w  C_  T  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
126, 11syl 16 . . . . . . . 8  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
1312rexlimdva 2832 . . . . . . 7  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T )  ->  ( E. w  e.  T  ~P z  C_  w  ->  ~P z  e.  T ) )
142, 13ralimdaa 2785 . . . . . 6  |-  ( A. z  e.  T  ~P z  C_  T  ->  ( A. z  e.  T  E. w  e.  T  ~P z  C_  w  ->  A. z  e.  T  ~P z  e.  T
) )
1514imdistani 673 . . . . 5  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  A. z  e.  T  E. w  e.  T  ~P z  C_  w )  -> 
( A. z  e.  T  ~P z  C_  T  /\  A. z  e.  T  ~P z  e.  T ) )
16 r19.26 2840 . . . . 5  |-  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  <->  ( A. z  e.  T  ~P z  C_  T  /\  A. z  e.  T  E. w  e.  T  ~P z  C_  w ) )
17 r19.26 2840 . . . . 5  |-  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
)  <->  ( A. z  e.  T  ~P z  C_  T  /\  A. z  e.  T  ~P z  e.  T ) )
1815, 16, 173imtr4i 259 . . . 4  |-  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  ->  A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
) )
19 ssid 3369 . . . . . . 7  |-  ~P z  C_ 
~P z
20 sseq2 3372 . . . . . . . 8  |-  ( w  =  ~P z  -> 
( ~P z  C_  w 
<->  ~P z  C_  ~P z ) )
2120rspcev 3054 . . . . . . 7  |-  ( ( ~P z  e.  T  /\  ~P z  C_  ~P z )  ->  E. w  e.  T  ~P z  C_  w )
2219, 21mpan2 654 . . . . . 6  |-  ( ~P z  e.  T  ->  E. w  e.  T  ~P z  C_  w )
2322anim2i 554 . . . . 5  |-  ( ( ~P z  C_  T  /\  ~P z  e.  T
)  ->  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) )
2423ralimi 2783 . . . 4  |-  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
)  ->  A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) )
2518, 24impbii 182 . . 3  |-  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w )  <->  A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T )
)
2625anbi1i 678 . 2  |-  ( ( 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 ) )  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
)  /\  A. z  e.  ~P  T ( z 
~~  T  \/  z  e.  T ) ) )
271, 26syl6bb 254 1  |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T
)  /\  A. z  e.  ~P  T ( z 
~~  T  \/  z  e.  T ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214    ~~ cen 7108   Tarskictsk 8625
This theorem is referenced by:  tskpw  8630  0tsk  8632  inttsk  8651  inatsk  8655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-pow 4379
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-tsk 8626
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