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Theorem tsken 8634
Description: 3rd axiom of a Tarski's class. A subset of a Tarski's class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsken  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )

Proof of Theorem tsken
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpw2g 4366 . . 3  |-  ( T  e.  Tarski  ->  ( A  e. 
~P T  <->  A  C_  T
) )
21biimpar 473 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  A  e.  ~P T )
3 eltskg 8630 . . . . 5  |-  ( T  e.  Tarski  ->  ( T  e. 
Tarski 
<->  ( A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  /\  A. x  e.  ~P  T
( x  ~~  T  \/  x  e.  T
) ) ) )
43ibi 234 . . . 4  |-  ( T  e.  Tarski  ->  ( A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  /\  A. x  e.  ~P  T
( x  ~~  T  \/  x  e.  T
) ) )
54simprd 451 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T ) )
6 breq1 4218 . . . . 5  |-  ( x  =  A  ->  (
x  ~~  T  <->  A  ~~  T ) )
7 eleq1 2498 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7orbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( x  ~~  T  \/  x  e.  T
)  <->  ( A  ~~  T  \/  A  e.  T ) ) )
98rspccva 3053 . . 3  |-  ( ( A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T )  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
105, 9sylan 459 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
112, 10syldan 458 1  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   ~Pcpw 3801   class class class wbr 4215    ~~ cen 7109   Tarskictsk 8628
This theorem is referenced by:  tskssel  8637  inttsk  8654  r1tskina  8662  tskuni  8663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333
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 4216  df-tsk 8629
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