Metamath Proof Explorer


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 opprc ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )
3 1 2 nsyl5 ( ¬ 𝐴 ∈ V → ⟨ 𝐴 , 𝐵 ⟩ = ∅ )