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	2	anbi1i	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
4	1 3	bitri	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
5		df-ot	⊢ ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩
6	5	eleq1i	⊢ ( ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) )
7		df-3an	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
8	4 6 7	3bitr4i	⊢ ( ⟨ 𝐴 , 𝐵 , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )