Theorem opelvv

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypotheses	opelvv.1	$$⊢A\in \mathrm{V}$$
	opelvv.2	$$⊢B\in \mathrm{V}$$
Assertion	opelvv	$$⊢⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		opelvv.1	$$⊢A\in \mathrm{V}$$
2		opelvv.2	$$⊢B\in \mathrm{V}$$
3		opelxpi	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$
4	1 2 3	mp2an	$$⊢⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$