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 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴𝐷𝐵𝐸𝐶𝐹 ) )

Proof

Step Hyp Ref Expression
1 opelxp ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ∧ 𝐶𝐹 ) )
2 opelxp ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ↔ ( 𝐴𝐷𝐵𝐸 ) )
3 1 2 bianbi ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ( 𝐴𝐷𝐵𝐸 ) ∧ 𝐶𝐹 ) )
4 df-ot 𝐴 , 𝐵 , 𝐶 ⟩ = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶
5 4 eleq1i ( ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) )
6 df-3an ( ( 𝐴𝐷𝐵𝐸𝐶𝐹 ) ↔ ( ( 𝐴𝐷𝐵𝐸 ) ∧ 𝐶𝐹 ) )
7 3 5 6 3bitr4i ( ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴𝐷𝐵𝐸𝐶𝐹 ) )