Theorem tsk0 9585

Description: A nonempty Tarski class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)

Assertion

Ref	Expression
tsk0	|- ( ( T e. Tarski /\ T =/= (/) ) -> (/) e. T )

Proof of Theorem tsk0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( T =/= (/) <-> E. x x e. T )
2		0ss 3972	. . . . . 6 |- (/) C_ x
3		tskss 9580	. . . . . 6 |- ( ( T e. Tarski /\ x e. T /\ (/) C_ x ) -> (/) e. T )
4	2, 3	mp3an3 1413	. . . . 5 |- ( ( T e. Tarski /\ x e. T ) -> (/) e. T )
5	4	expcom 451	. . . 4 |- ( x e. T -> ( T e. Tarski -> (/) e. T ) )
6	5	exlimiv 1858	. . 3 |- ( E. x x e. T -> ( T e. Tarski -> (/) e. T ) )
7	1, 6	sylbi 207	. 2 |- ( T =/= (/) -> ( T e. Tarski -> (/) e. T ) )
8	7	impcom 446	1 |- ( ( T e. Tarski /\ T =/= (/) ) -> (/) e. T )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-br 4654  df-tsk 9571

This theorem is referenced by:  tsk1  9586  tskr1om  9589