Theorem 0tsk 9577

Description: The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)

Assertion

Ref	Expression
0tsk	|- (/) e. Tarski

Proof of Theorem 0tsk

Step	Hyp	Ref	Expression
1		ral0 4076	. 2 |- A. x e. (/) ( ~P x C_ (/) /\ ~P x e. (/) )
2		elsni 4194	. . . . 5 |- ( x e. { (/) } -> x = (/) )
3		0ex 4790	. . . . . . . 8 |- (/) e. _V
4	3	enref 7988	. . . . . . 7 |- (/) ~~ (/)
5		breq1 4656	. . . . . . 7 |- ( x = (/) -> ( x ~~ (/) <-> (/) ~~ (/) ) )
6	4, 5	mpbiri 248	. . . . . 6 |- ( x = (/) -> x ~~ (/) )
7	6	orcd 407	. . . . 5 |- ( x = (/) -> ( x ~~ (/) \/ x e. (/) ) )
8	2, 7	syl 17	. . . 4 |- ( x e. { (/) } -> ( x ~~ (/) \/ x e. (/) ) )
9		pw0 4343	. . . 4 |- ~P (/) = { (/) }
10	8, 9	eleq2s 2719	. . 3 |- ( x e. ~P (/) -> ( x ~~ (/) \/ x e. (/) ) )
11	10	rgen 2922	. 2 |- A. x e. ~P (/) ( x ~~ (/) \/ x e. (/) )
12		eltsk2g 9573	. . 3 |- ( (/) e. _V -> ( (/) e. Tarski <-> ( A. x e. (/) ( ~P x C_ (/) /\ ~P x e. (/) ) /\ A. x e. ~P (/) ( x ~~ (/) \/ x e. (/) ) ) ) )
13	3, 12	ax-mp 5	. 2 |- ( (/) e. Tarski <-> ( A. x e. (/) ( ~P x C_ (/) /\ ~P x e. (/) ) /\ A. x e. ~P (/) ( x ~~ (/) \/ x e. (/) ) ) )
14	1, 11, 13	mpbir2an 955	1 |- (/) e. Tarski

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   ~~ cen 7952   Tarskictsk 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-en 7956  df-tsk 9571

This theorem is referenced by:  r1tskina  9604  grutsk  9644  tskmcl  9663