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	$$⊢A\in C$$
	opelxpii.2	$$⊢B\in D$$
Assertion	opelxpii	$$⊢⟨A,B⟩\in \left(C\times D\right)$$

Proof

Step	Hyp	Ref	Expression
1		opelxpii.1	$$⊢A\in C$$
2		opelxpii.2	$$⊢B\in D$$
3		opelxpi	$$⊢\left(A\in C\wedge B\in D\right)\to ⟨A,B⟩\in \left(C\times D\right)$$
4	1 2 3	mp2an	$$⊢⟨A,B⟩\in \left(C\times D\right)$$