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Theorem eltsk2g 9176
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 9175 . 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 2769 . . . . . . 7  |-  F/ z A. z  e.  T  ~P z  C_  T
3 pweq 3954 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ~P z  =  ~P w
)
43sseq1d 3459 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( ~P z  C_  T  <->  ~P w  C_  T ) )
54rspccva 3149 . . . . . . . . . 10  |-  ( ( A. z  e.  T  ~P z  C_  T  /\  w  e.  T )  ->  ~P w  C_  T
)
65adantlr 721 . . . . . . . . 9  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ~P w  C_  T )
7 vex 3048 . . . . . . . . . . . 12  |-  z  e. 
_V
87pwex 4586 . . . . . . . . . . 11  |-  ~P z  e.  _V
98elpw 3957 . . . . . . . . . 10  |-  ( ~P z  e.  ~P w  <->  ~P z  C_  w )
10 ssel 3426 . . . . . . . . . 10  |-  ( ~P w  C_  T  ->  ( ~P z  e.  ~P w  ->  ~P z  e.  T ) )
119, 10syl5bir 222 . . . . . . . . 9  |-  ( ~P w  C_  T  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
126, 11syl 17 . . . . . . . 8  |-  ( ( ( A. z  e.  T  ~P z  C_  T  /\  z  e.  T
)  /\  w  e.  T )  ->  ( ~P z  C_  w  ->  ~P z  e.  T
) )
1312rexlimdva 2879 . . . . . . 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 2790 . . . . . 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 696 . . . . 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 2917 . . . . 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 2917 . . . . 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 270 . . . 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 3451 . . . . . . 7  |-  ~P z  C_ 
~P z
20 sseq2 3454 . . . . . . . 8  |-  ( w  =  ~P z  -> 
( ~P z  C_  w 
<->  ~P z  C_  ~P z ) )
2120rspcev 3150 . . . . . . 7  |-  ( ( ~P z  e.  T  /\  ~P z  C_  ~P z )  ->  E. w  e.  T  ~P z  C_  w )
2219, 21mpan2 677 . . . . . 6  |-  ( ~P z  e.  T  ->  E. w  e.  T  ~P z  C_  w )
2322anim2i 573 . . . . 5  |-  ( ( ~P z  C_  T  /\  ~P z  e.  T
)  ->  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) )
2423ralimi 2781 . . . 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 191 . . 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 701 . 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 265 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 188    \/ wo 370    /\ wa 371    e. wcel 1887   A.wral 2737   E.wrex 2738    C_ wss 3404   ~Pcpw 3951   class class class wbr 4402    ~~ cen 7566   Tarskictsk 9173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-pow 4581
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-tsk 9174
This theorem is referenced by:  tskpw  9178  0tsk  9180  inttsk  9199  inatsk  9203
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