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Theorem eltsk2g 8389
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 8388 . 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 2606 . . . . . . 7  |-  F/ z A. z  e.  T  ~P z  C_  T
3 pweq 3641 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ~P z  =  ~P w
)
43sseq1d 3218 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( ~P z  C_  T  <->  ~P w  C_  T ) )
54rspccva 2896 . . . . . . . . . 10  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  w  e.  T )  ->  ~P w  C_  T
)
65adantlr 695 . . . . . . . . 9  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ~P w  C_  T )
7 vex 2804 . . . . . . . . . . . 12  |-  z  e. 
_V
87pwex 4209 . . . . . . . . . . 11  |-  ~P z  e.  _V
98elpw 3644 . . . . . . . . . 10  |-  ( ~P z  e.  ~P w  <->  ~P z  C_  w )
10 ssel 3187 . . . . . . . . . 10  |-  ( ~P w  C_  T  ->  ( ~P z  e.  ~P w  ->  ~P z  e.  T ) )
119, 10syl5bir 209 . . . . . . . . 9  |-  ( ~P w  C_  T  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
126, 11syl 15 . . . . . . . 8  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
1312rexlimdva 2680 . . . . . . 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 2633 . . . . . 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 671 . . . . 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 2688 . . . . 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 2688 . . . . 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 257 . . . 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 3210 . . . . . . 7  |-  ~P z  C_ 
~P z
20 sseq2 3213 . . . . . . . 8  |-  ( w  =  ~P z  -> 
( ~P z  C_  w 
<->  ~P z  C_  ~P z ) )
2120rspcev 2897 . . . . . . 7  |-  ( ( ~P z  e.  T  /\  ~P z  C_  ~P z )  ->  E. w  e.  T  ~P z  C_  w )
2219, 21mpan2 652 . . . . . 6  |-  ( ~P z  e.  T  ->  E. w  e.  T  ~P z  C_  w )
2322anim2i 552 . . . . 5  |-  ( ( ~P z  C_  T  /\  ~P z  e.  T
)  ->  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) )
2423ralimi 2631 . . . 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 180 . . 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 676 . 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 252 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    ~~ cen 6876   Tarskictsk 8386
This theorem is referenced by:  tskpw  8391  0tsk  8393  inttsk  8412  inatsk  8416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-tsk 8387
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