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 ∈ V
	opth1.2	⊢ 𝐵 ∈ V
Assertion	opnzi	⊢ ⟨ 𝐴 , 𝐵 ⟩ ≠ ∅

Proof

Step	Hyp	Ref	Expression
1		opth1.1	⊢ 𝐴 ∈ V
2		opth1.2	⊢ 𝐵 ∈ V
3		opnz	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ≠ ∅ ↔ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
4	1 2 3	mpbir2an	⊢ ⟨ 𝐴 , 𝐵 ⟩ ≠ ∅