Theorem otelxp

Description: Ordered triple membership in a triple Cartesian product. (Contributed by Scott Fenton, 31-Jan-2025)

	Ref	Expression
Assertion	otelxp	|- ( <. 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 >. , C >. e. ( ( D X. E ) X. F ) <-> ( <. A , B >. e. ( D X. E ) /\ C e. F ) )
2		opelxp	|- ( <. A , B >. e. ( D X. E ) <-> ( A e. D /\ B e. E ) )
3	2	anbi1i	|- ( ( <. A , B >. e. ( D X. E ) /\ C e. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
4	1 3	bitri	|- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
5		df-ot	|- <. A , B , C >. = <. <. A , B >. , C >.
6	5	eleq1i	|- ( <. A , B , C >. e. ( ( D X. E ) X. F ) <-> <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) )
7		df-3an	|- ( ( A e. D /\ B e. E /\ C e. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
8	4 6 7	3bitr4i	|- ( <. A , B , C >. e. ( ( D X. E ) X. F ) <-> ( A e. D /\ B e. E /\ C e. F ) )