Theorem tsken

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)

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

Proof

Step	Hyp	Ref	Expression
1		eltskg	|- ( 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	|- ( 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	|- ( T e. Tarski -> A. x e. ~P T ( x ~~ T \/ x e. T ) )
4		elpw2g	|- ( T e. Tarski -> ( A e. ~P T <-> A C_ T ) )
5	4	biimpar	|- ( ( T e. Tarski /\ A C_ T ) -> A e. ~P T )
6		breq1	|- ( x = A -> ( x ~~ T <-> A ~~ T ) )
7		eleq1	|- ( x = A -> ( x e. T <-> A e. T ) )
8	6 7	orbi12d	|- ( x = A -> ( ( x ~~ T \/ x e. T ) <-> ( A ~~ T \/ A e. T ) ) )
9	8	rspccva	|- ( ( A. x e. ~P T ( x ~~ T \/ x e. T ) /\ A e. ~P T ) -> ( A ~~ T \/ A e. T ) )
10	3 5 9	syl2an2r	|- ( ( T e. Tarski /\ A C_ T ) -> ( A ~~ T \/ A e. T ) )