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Theorem 0tsk 8393
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 3571 . 2  |-  A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )
2 elsni 3677 . . . . 5  |-  ( x  e.  { (/) }  ->  x  =  (/) )
3 0ex 4166 . . . . . . . 8  |-  (/)  e.  _V
43enref 6910 . . . . . . 7  |-  (/)  ~~  (/)
5 breq1 4042 . . . . . . 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 3778 . . . 4  |-  ~P (/)  =  { (/)
}
108, 9eleq2s 2388 . . 3  |-  ( x  e.  ~P (/)  ->  (
x  ~~  (/)  \/  x  e.  (/) ) )
1110rgen 2621 . 2  |-  A. x  e.  ~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) )
12 eltsk2g 8389 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039    ~~ cen 6876   Tarskictsk 8386
This theorem is referenced by:  r1tskina  8420  grutsk  8460  tskmcl  8479
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  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-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6880  df-tsk 8387
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