Theorem opnzi

Description: An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypotheses	opth1.1	$$⊢A\in \mathrm{V}$$
	opth1.2	$$⊢B\in \mathrm{V}$$
Assertion	opnzi	$$⊢⟨A,B⟩\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		opth1.1	$$⊢A\in \mathrm{V}$$
2		opth1.2	$$⊢B\in \mathrm{V}$$
3		opnz	$$⊢⟨A,B⟩\ne \varnothing \leftrightarrow \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$
4	1 2 3	mpbir2an	$$⊢⟨A,B⟩\ne \varnothing$$