Theorem opelvvg

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015)

	Ref	Expression
Assertion	opelvvg	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in V\to A\in \mathrm{V}$$
2		elex	$$⊢B\in W\to B\in \mathrm{V}$$
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	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$