Theorem ot2elxp

Description: Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024)

	Ref	Expression
Assertion	ot2elxp	|- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> ( A e. D /\ B e. E /\ C e. F ) )

Proof

Step	Hyp	Ref	Expression
1		opelxp	|- ( <. A , B >. e. ( D X. E ) <-> ( A e. D /\ B e. E ) )
2	1	anbi1i	|- ( ( <. A , B >. e. ( D X. E ) /\ C e. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
3		opelxp	|- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> ( <. A , B >. e. ( D X. E ) /\ C e. F ) )
4		df-3an	|- ( ( A e. D /\ B e. E /\ C e. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
5	2 3 4	3bitr4i	|- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> ( A e. D /\ B e. E /\ C e. F ) )