Theorem tskop

Description: If A and B are members of a Tarski class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011)

	Ref	Expression
Assertion	tskop	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> <. A , B >. e. T )

Proof

Step	Hyp	Ref	Expression
1		dfopg	|- ( ( A e. T /\ B e. T ) -> <. A , B >. = { { A } , { A , B } } )
2	1	3adant1	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> <. A , B >. = { { A } , { A , B } } )
3		simp1	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> T e. Tarski )
4		tsksn	|- ( ( T e. Tarski /\ A e. T ) -> { A } e. T )
5	4	3adant3	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> { A } e. T )
6		tskpr	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> { A , B } e. T )
7		tskpr	|- ( ( T e. Tarski /\ { A } e. T /\ { A , B } e. T ) -> { { A } , { A , B } } e. T )
8	3 5 6 7	syl3anc	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> { { A } , { A , B } } e. T )
9	2 8	eqeltrd	|- ( ( T e. Tarski /\ A e. T /\ B e. T ) -> <. A , B >. e. T )