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Theorem 0tsk 8377
Description: The empty set is a (transitive) Tarski's class. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
0tsk  |-  (/)  e.  Tarski

Proof of Theorem 0tsk
StepHypRef Expression
1 ral0 3558 . 2  |-  A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )
2 elsni 3664 . . . . 5  |-  ( x  e.  { (/) }  ->  x  =  (/) )
3 0ex 4150 . . . . . . . 8  |-  (/)  e.  _V
43enref 6894 . . . . . . 7  |-  (/)  ~~  (/)
5 breq1 4026 . . . . . . 7  |-  ( x  =  (/)  ->  ( x 
~~  (/)  <->  (/)  ~~  (/) ) )
64, 5mpbiri 224 . . . . . 6  |-  ( x  =  (/)  ->  x  ~~  (/) )
76orcd 381 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~~  (/)  \/  x  e.  (/) ) )
82, 7syl 15 . . . 4  |-  ( x  e.  { (/) }  ->  ( x  ~~  (/)  \/  x  e.  (/) ) )
9 pw0 3762 . . . 4  |-  ~P (/)  =  { (/)
}
108, 9eleq2s 2375 . . 3  |-  ( x  e.  ~P (/)  ->  (
x  ~~  (/)  \/  x  e.  (/) ) )
1110rgen 2608 . 2  |-  A. x  e.  ~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) )
12 eltsk2g 8373 . . 3  |-  ( (/)  e.  _V  ->  ( (/)  e.  Tarski  <->  ( A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )  /\  A. x  e.  ~P  (/) ( x 
~~  (/)  \/  x  e.  (/) ) ) ) )
133, 12ax-mp 8 . 2  |-  ( (/)  e.  Tarski 
<->  ( A. x  e.  (/)  ( ~P x  C_  (/) 
/\  ~P x  e.  (/) )  /\  A. x  e. 
~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) ) ) )
141, 11, 13mpbir2an 886 1  |-  (/)  e.  Tarski
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023    ~~ cen 6860   Tarskictsk 8370
This theorem is referenced by:  r1tskina  8404  grutsk  8444  tskmcl  8463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864  df-tsk 8371
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