Theorem tsken 9576

Description: Third axiom of a Tarski class. A subset of a Tarski class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tsken	|- ( ( T e. Tarski /\ A C_ T ) -> ( A ~~ T \/ A e. T ) )

Proof of Theorem tsken

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltskg 9572	. . . 4 |- ( T e. Tarski -> ( T e. Tarski <-> ( A. x e. T ( ~P x C_ T /\ E. y e. T ~P x C_ y ) /\ A. x e. ~P T ( x ~~ T \/ x e. T ) ) ) )
2	1	ibi 256	. . 3 |- ( T e. Tarski -> ( A. x e. T ( ~P x C_ T /\ E. y e. T ~P x C_ y ) /\ A. x e. ~P T ( x ~~ T \/ x e. T ) ) )
3	2	simprd 479	. 2 |- ( T e. Tarski -> A. x e. ~P T ( x ~~ T \/ x e. T ) )
4		elpw2g 4827	. . 3 |- ( T e. Tarski -> ( A e. ~P T <-> A C_ T ) )
5	4	biimpar 502	. 2 |- ( ( T e. Tarski /\ A C_ T ) -> A e. ~P T )
6		breq1 4656	. . . 4 |- ( x = A -> ( x ~~ T <-> A ~~ T ) )
7		eleq1 2689	. . . 4 |- ( x = A -> ( x e. T <-> A e. T ) )
8	6, 7	orbi12d 746	. . 3 |- ( x = A -> ( ( x ~~ T \/ x e. T ) <-> ( A ~~ T \/ A e. T ) ) )
9	8	rspccva 3308	. 2 |- ( ( A. x e. ~P T ( x ~~ T \/ x e. T ) /\ A e. ~P T ) -> ( A ~~ T \/ A e. T ) )
10	3, 5, 9	syl2an2r 876	1 |- ( ( T e. Tarski /\ A C_ T ) -> ( A ~~ T \/ A e. T ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   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-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-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:  tskssel  9579  inttsk  9596  r1tskina  9604  tskuni  9605