Theorem ot2elxp

Description: Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024)

	Ref	Expression
Assertion	ot2elxp	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		opelxp	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) )
2	1	anbi1i	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
3		opelxp	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐷 × 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
4		df-3an	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ) ∧ 𝐶 ∈ 𝐹 ) )
5	2 3 4	3bitr4i	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ ( ( 𝐷 × 𝐸 ) × 𝐹 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹 ) )