Theorem opelxpii

Description: Ordered pair membership in a Cartesian product (implication), induction form. (Contributed by Steven Nguyen, 17-Jul-2022)

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

Proof

Step	Hyp	Ref	Expression
1		opelxpii.1	⊢ 𝐴 ∈ 𝐶
2		opelxpii.2	⊢ 𝐵 ∈ 𝐷
3		opelxpi	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )
4	1 2 3	mp2an	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 )