Theorem opelxpd

Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	opelxpd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
	opelxpd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
Assertion	opelxpd	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		opelxpd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2		opelxpd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
3		opelxpi	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )