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Metamath Proof Explorer

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 )