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Theorem eltsk2g 9125
Description: Properties of a Tarski 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 9124 . 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 2845 . . . . . . 7  |-  F/ z A. z  e.  T  ~P z  C_  T
3 pweq 4013 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ~P z  =  ~P w
)
43sseq1d 3531 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( ~P z  C_  T  <->  ~P w  C_  T ) )
54rspccva 3213 . . . . . . . . . 10  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  w  e.  T )  ->  ~P w  C_  T
)
65adantlr 714 . . . . . . . . 9  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ~P w  C_  T )
7 vex 3116 . . . . . . . . . . . 12  |-  z  e. 
_V
87pwex 4630 . . . . . . . . . . 11  |-  ~P z  e.  _V
98elpw 4016 . . . . . . . . . 10  |-  ( ~P z  e.  ~P w  <->  ~P z  C_  w )
10 ssel 3498 . . . . . . . . . 10  |-  ( ~P w  C_  T  ->  ( ~P z  e.  ~P w  ->  ~P z  e.  T ) )
119, 10syl5bir 218 . . . . . . . . 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 2955 . . . . . . 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 2866 . . . . . 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 690 . . . . 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 2989 . . . . 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 2989 . . . . 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 266 . . . 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 3523 . . . . . . 7  |-  ~P z  C_ 
~P z
20 sseq2 3526 . . . . . . . 8  |-  ( w  =  ~P z  -> 
( ~P z  C_  w 
<->  ~P z  C_  ~P z ) )
2120rspcev 3214 . . . . . . 7  |-  ( ( ~P z  e.  T  /\  ~P z  C_  ~P z )  ->  E. w  e.  T  ~P z  C_  w )
2219, 21mpan2 671 . . . . . 6  |-  ( ~P z  e.  T  ->  E. w  e.  T  ~P z  C_  w )
2322anim2i 569 . . . . 5  |-  ( ( ~P z  C_  T  /\  ~P z  e.  T
)  ->  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) )
2423ralimi 2857 . . . 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 188 . . 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 695 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-pow 4625
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:  tskpw  9127  0tsk  9129  inttsk  9148  inatsk  9152
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