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Theorem 0tsk 8630
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 3732 . 2  |-  A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )
2 elsni 3838 . . . . 5  |-  ( x  e.  { (/) }  ->  x  =  (/) )
3 0ex 4339 . . . . . . . 8  |-  (/)  e.  _V
43enref 7140 . . . . . . 7  |-  (/)  ~~  (/)
5 breq1 4215 . . . . . . 7  |-  ( x  =  (/)  ->  ( x 
~~  (/)  <->  (/)  ~~  (/) ) )
64, 5mpbiri 225 . . . . . 6  |-  ( x  =  (/)  ->  x  ~~  (/) )
76orcd 382 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~~  (/)  \/  x  e.  (/) ) )
82, 7syl 16 . . . 4  |-  ( x  e.  { (/) }  ->  ( x  ~~  (/)  \/  x  e.  (/) ) )
9 pw0 3945 . . . 4  |-  ~P (/)  =  { (/)
}
108, 9eleq2s 2528 . . 3  |-  ( x  e.  ~P (/)  ->  (
x  ~~  (/)  \/  x  e.  (/) ) )
1110rgen 2771 . 2  |-  A. x  e.  ~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) )
12 eltsk2g 8626 . . 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 887 1  |-  (/)  e.  Tarski
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   class class class wbr 4212    ~~ cen 7106   Tarskictsk 8623
This theorem is referenced by:  r1tskina  8657  grutsk  8697  tskmcl  8716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-en 7110  df-tsk 8624
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