Theorem opprc1

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc . (Contributed by NM, 10-Apr-2004) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opprc1	⊢ ( ¬ 𝐴 ∈ V → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐴 ∈ V )
2	1	con3i	⊢ ( ¬ 𝐴 ∈ V → ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
3		opprc	⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )
4	2 3	syl	⊢ ( ¬ 𝐴 ∈ V → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )