This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 1 2 bianbi 
 |-  ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
4 df-ot 
 |-  <. A , B , C >. = <. <. A , B >. , C >.
5 4 eleq1i 
 |-  ( <. A , B , C >. e. ( ( D X. E ) X. F ) <-> <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) )
6 df-3an 
 |-  ( ( A e. D /\ B e. E /\ C e. F ) <-> ( ( A e. D /\ B e. E ) /\ C e. F ) )
7 3 5 6 3bitr4i 
 |-  ( <. A , B , C >. e. ( ( D X. E ) X. F ) <-> ( A e. D /\ B e. E /\ C e. F ) )