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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 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 opelxp ⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
2 opelxp ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) )
3 1 2 bianbi ⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
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 ⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
7 3 5 6 3bitr4i ⊢ ( ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )